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- 2 \\sin^2 \\theta)$ for $\\cos 2\\theta$ and simplifies to the required form A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Applies the compound angle formula to expand sin(2θ + θ).","code":"M"},{"id":"p2","marks":1,"description":"Uses double angle identities for sin 2θ and cos 2θ and simplifies to 3sinθ - 4sin³θ.","code":"A"}],"finalAnswers":["3\\sin\\theta - 4\\sin^3\\theta"],"generatedAt":"2026-05-06T14:04:27.271Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"trigonometric_identities","label":"Using Trigonometric Identities","steps":[{"type":"text","order":0,"title":"Apply addition formula","content":"To express $\\sin 3\\theta$ in terms of $\\sin \\theta$, we start by using the addition formula (also known as the compound angle identity) for sine:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nApplying this identity to $\\sin(2\\theta + \\theta)$, we get:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos \\theta + \\cos 2\\theta \\sin \\theta$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$${\\sin 3\\theta} = \\htmlData{anchor=sin2t-eq}{\\color{@sky}{\\sin 2\\theta}} \\cos \\theta + \\htmlData{anchor=cos2t-eq}{\\color{@amber}{\\cos 2\\theta}} \\sin \\theta$"},{"latex":"$${\\sin 3\\theta} = \\htmlData{anchor=sin2t-sub}{\\color{@sky}{(2\\sin\\theta\\cos\\theta)}} \\cos\\theta + \\htmlData{anchor=cos2t-sub}{\\color{@amber}{(1 - 2\\sin^2\\theta)}} \\sin\\theta$"},{"text":"Next, we replace the double-angle terms using the standard double-angle formulas:"},{"text":"- $\\sin 2\\theta = 2\\sin\\theta\\cos\\theta$"},{"text":"- $\\cos 2\\theta = 1 - 2\\sin^2\\theta$ (choosing this version to work directly towards terms containing sine)"}],"highlights":[{"color":"sky","style":"underline","anchor":"sin2t-eq"},{"color":"amber","style":"underline","anchor":"cos2t-eq"}],"connections":[{"to":"sin2t-sub","from":"sin2t-eq","color":"sky","label":"Substitute $2\\sin\\theta\\cos\\theta$","style":"solid"},{"to":"cos2t-sub","from":"cos2t-eq","color":"amber","label":"Substitute $1 - 2\\sin^2\\theta$","style":"solid"}]},"order":1,"title":"Substitute double-angle formulas"},{"type":"text","order":2,"title":"Simplify using identity","content":"Now we simplify the expression:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta\\cos^2\\theta + \\sin\\theta(1 - 2\\sin^2\\theta) \\\\ &= 2\\sin\\theta\\cos^2\\theta + \\sin\\theta - 2\\sin^3\\theta \\end{aligned}$$\n\nTo make the entire equation in terms of $\\sin\\theta$, we substitute the Pythagorean identity $\\cos^2\\theta = 1 - \\sin^2\\theta$ into the first term:\n\n$$\\sin 3\\theta = 2\\sin\\theta(1 - \\sin^2\\theta) + \\sin\\theta - 2\\sin^3\\theta$$"},{"type":"text","order":3,"title":"Expand and collect terms","content":"Expand the first term and group all the like terms together:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta - 2\\sin^3\\theta + \\sin\\theta - 2\\sin^3\\theta \\\\ &= (2\\sin\\theta + \\sin\\theta) + (-2\\sin^3\\theta - 2\\sin^3\\theta) \\\\ &= 3\\sin\\theta - 4\\sin^3\\theta \\end{aligned}$$\n\nThis completes the proof for Part 1."},{"type":"text","order":4,"title":"Identify the identity","content":"For Part 2, we need to solve:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.5$$\n\nUsing the identity we proved in Part 1, we can rewrite the left-hand side of the equation as $\\sin 3\\theta$:\n\n$$\\sin 3\\theta = 0.5$$"},{"type":"text","order":5,"title":"Find the general solutions","content":"We are given the interval $0 \\leq \\theta \\leq \\pi$. \n\nMultiplying the inequality by $3$, we find the corresponding range for $3\\theta$:\n\n$$0 \\leq 3\\theta \\leq 3\\pi$$\n\nWithin the interval $[0, 3\\pi]$, we find the values where the sine function is equal to $0.5$:\n\n$$\\begin{aligned} 3\\theta &= \\frac{\\pi}{6}, \\pi - \\frac{\\pi}{6}, 2\\pi + \\frac{\\pi}{6}, 3\\pi - \\frac{\\pi}{6} \\\\ 3\\theta &= \\frac{\\pi}{6}, \\frac{5\\pi}{6}, \\frac{13\\pi}{6}, \\frac{17\\pi}{6} \\end{aligned}$$"},{"type":"text","order":6,"title":"Solve for theta","content":"To find $\\theta$, we divide each value by $3$:\n\n$$\\theta = \\frac{\\pi}{18}, \\frac{5\\pi}{18}, \\frac{13\\pi}{18}, \\frac{17\\pi}{18}$$\n\nThese four solutions all lie within the given range $0 \\leq \\theta \\leq \\pi$."}]}],"generatedAt":"2026-06-09T05:03:54.942Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"7f84acd4-2695-49e2-969b-a71a991e5694","label":"b","content":"Solve, for $0 \\leq \\theta \\leq \\pi$, the equation,\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.5$$\n\nGive your answers in terms of $\\pi$.","markscheme":"- Recognises the LHS as $\\sin 3\\theta$, so $\\sin 3\\theta = 0.5$ M1\n- Finds values for $3\\theta$ in the range $[0, 3\\pi]$: $\\frac{\\pi}{6}, \\frac{5\\pi}{6}, \\frac{13\\pi}{6}, \\frac{17\\pi}{6}$ M1\n- Divides by 3 to find $\\theta$ values M1\n- Final answers: $\\frac{1}{18}\\pi, \\frac{5}{18}\\pi, \\frac{13}{18}\\pi, \\frac{17}{18}\\pi$ A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Identifies that the equation is sin 3θ = 0.5.","code":"M"},{"id":"p2","marks":1,"description":"Finds at least three correct values for 3θ within the range [0, 3π].","code":"M"},{"id":"p3","marks":1,"description":"Correctly divides all 3θ values by 3 to obtain values for θ.","code":"M"},{"id":"p4","marks":1,"description":"Provides the final correct set of solutions in terms of π.","code":"A"}],"finalAnswers":["\\frac{1}{18}\\pi, \\frac{5}{18}\\pi, \\frac{13}{18}\\pi, \\frac{17}{18}\\pi"],"generatedAt":"2026-05-06T14:04:27.271Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"triple_angle_identity","label":"Triple Angle Identity","steps":[{"type":"text","order":0,"title":"Prove the identity","content":"For Part 1, we show that $\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$ by expanding $\\sin(2\\theta + \\theta)$ using the sine compound angle identity:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nLetting $A = 2\\theta$ and $B = \\theta$, we get:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos \\theta + \\cos 2\\theta \\sin \\theta$$\n\nNext, we use the double-angle formulas $\\sin 2\\theta = 2\\sin\\theta\\cos\\theta$ and $\\cos 2\\theta = 1 - 2\\sin^2\\theta$ to substitute:\n\n$$\\sin 3\\theta = (2\\sin\\theta\\cos\\theta)\\cos\\theta + (1 - 2\\sin^2\\theta)\\sin\\theta$$\n\n$$\\sin 3\\theta = 2\\sin\\theta\\cos^2\\theta + \\sin\\theta - 2\\sin^3\\theta$$\n\nUsing the fundamental identity $\\cos^2\\theta = 1 - \\sin^2\\theta$, we express everything in terms of $\\sin\\theta$:\n\n$$\\sin 3\\theta = 2\\sin\\theta(1 - \\sin^2\\theta) + \\sin\\theta - 2\\sin^3\\theta$$\n\n$$\\sin 3\\theta = 2\\sin\\theta - 2\\sin^3\\theta + \\sin\\theta - 2\\sin^3\\theta$$\n\n$$\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$$\n\nThis completes the proof for Part 1."},{"type":"text","order":1,"title":"Recognise triple angle identity","content":"For Part 2, the left-hand side of the given equation is $3\\sin\\theta - 4\\sin^3\\theta$. We recognise this as the triple-angle identity we proved in Part 1:\n\n$$\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$$\n\nSubstituting this identity into our equation, we can rewrite it in a much simpler form:\n\n$$\\sin 3\\theta = 0.5$$"},{"type":"text","order":2,"title":"Find the new interval","content":"We are given the interval for $\\theta$ as:\n\n$$0 \\leq \\theta \\leq \\pi$$\n\nSince our equation is now in terms of $3\\theta$, we must multiply the entire inequality by $3$ to find the corresponding interval for $3\\theta$:\n\n$$0 \\leq 3\\theta \\leq 3\\pi$$"},{"type":"text","order":3,"title":"Solve for 3theta","content":"Now we find all solutions to $\\sin 3\\theta = 0.5$ within the interval $[0, 3\\pi]$.\n\nThe principal value is:\n\n$$3\\theta = \\arcsin(0.5) = \\frac{\\pi}{6}$$\n\nUsing the symmetry of the sine graph, the second solution is:\n\n$$3\\theta = \\pi - \\frac{\\pi}{6} = \\frac{5\\pi}{6}$$\n\nTo find the rest of the solutions within the range $[0, 3\\pi]$, we add the period $2\\pi$ to each of our first two solutions:\n\n$$3\\theta = \\frac{\\pi}{6} + 2\\pi = \\frac{13\\pi}{6}$$\n\n$$3\\theta = \\frac{5\\pi}{6} + 2\\pi = \\frac{17\\pi}{6}$$\n\nThese four values lie within $[0, 3\\pi]$:\n\n$$3\\theta = \\frac{\\pi}{6}, \\; \\frac{5\\pi}{6}, \\; \\frac{13\\pi}{6}, \\; \\frac{17\\pi}{6}$$"},{"type":"text","order":4,"title":"Solve for theta","content":"To find the values of $\\theta$, we divide each of our solutions for $3\\theta$ by $3$:\n\n$$\\theta = \\frac{\\pi}{18}, \\; \\frac{5\\pi}{18}, \\; \\frac{13\\pi}{18}, \\; \\frac{17\\pi}{18}$$\n\nEach of these values lies within the original interval $[0, \\pi]$, so they are all valid solutions."}]},{"id":"graphing_calculator","label":"Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the functions","content":"To solve the equation using a graphing calculator, we first set up two functions to graph. We will look for their intersection points within the given interval:\n\n$$y_1 = 3\\sin\\theta - 4\\sin^3\\theta$$\n$$y_2 = 0.5$$\n\nWe restrict our view to the domain $0 \\leq \\theta \\leq \\pi$. On your calculator, use the variable $x$ instead of $\\theta$."},{"type":"text","order":1,"title":"Graph and find intersections","content":"Set your calculator's angle mode to **Radians**. \n\nEnter $y_1 = 3\\sin(x) - 4(\\sin(x))^3$ and $y_2 = 0.5$, then adjust your window settings so you can clearly see the intersections from $x = 0$ to $x = \\pi$ (approximately $0$ to $3.1416$). \n\nUse the intersection tool to find the $x$-coordinates of the four points where the curves cross.","calculator":[{"gdc":{"expression":"y = 3*sin(x) - 4*(sin(x))^3","instructions":"1. Go to Graph mode and set your angle unit to Radians.\n2. Enter the functions:\n $f1(x) = 3\\sin(x) - 4(\\sin(x))^3$\n $f2(x) = 0.5$\n3. Adjust the view window so that $x$ ranges from 0 to $\\pi$.\n4. Use the G-Solv or Analyze Graph tool to find the intersections."},"ti84":{"expression":"y = 3*sin(x) - 4*(sin(x))^3","instructions":"1. Press [MODE], ensure 'RADIAN' is highlighted.\n2. Press [Y=] and enter:\n $Y_1 = 3\\sin(X) - 4(\\sin(X))^3$\n $Y_2 = 0.5$\n3. Press [WINDOW] and set: \n $X_{min} = 0$, $X_{max} = \\pi$, $Y_{min} = -1.5$, $Y_{max} = 1.5$.\n4. Press [GRAPH].\n5. Press [2nd] [CALC] and select '5: intersect'. Select the first curve, second curve, and guess near each intersection point to find all four coordinates."},"desmos":{"expression":"y = 3*sin(x) - 4*(\\sin(x))^3 \\{0 \\le x \\le \\pi\\}","instructions":"1. Ensure the graph is in Radians mode (click the wrench icon in the top right).\n2. Type the equations:\n $y = 3\\sin(x) - 4(\\sin(x))^3$\n $y = 0.5$\n3. Restrict the domain of the first function by typing $\\{0 \\le x \\le \\pi\\}$.\n4. Click on the gray intersection points to reveal their coordinates."},"description":"Graph the functions and find the intersection points in Radians mode."}]},{"type":"text","order":2,"title":"Record the decimal values","content":"From the graph, we find four intersection points in the interval $[0, \\pi]$. The approximate $x$-values (representing $\\theta$) are:\n\n* $\\theta_1 \\approx 0.17453$\n* $\\theta_2 \\approx 0.87266$\n* $\\theta_3 \\approx 2.26893$\n* $\\theta_4 \\approx 2.96706$"},{"type":"text","order":3,"title":"Convert to exact multiples","content":"To write these decimal values in terms of $\\pi$ as requested by the question, divide each value by $\\pi$ to find the corresponding fractional coefficient:\n\n$$\\frac{0.17453}{\\pi} \\approx 0.05556 \\approx \\frac{1}{18} \\implies \\theta_1 = \\frac{1}{18}\\pi$$\n\n$$\\frac{0.87266}{\\pi} \\approx 0.27778 \\approx \\frac{5}{18} \\implies \\theta_2 = \\frac{5}{18}\\pi$$\n\n$$\\frac{2.26893}{\\pi} \\approx 0.72222 \\approx \\frac{13}{18} \\implies \\theta_3 = \\frac{13}{18}\\pi$$\n\n$$\\frac{2.96706}{\\pi} \\approx 0.94444 \\approx \\frac{17}{18} \\implies \\theta_4 = \\frac{17}{18}\\pi$$\n\nThus, the four solutions in the interval $0 \\leq \\theta \\leq \\pi$ are:\n\n$$\\theta = \\frac{1}{18}\\pi, \\; \\frac{5}{18}\\pi, \\; \\frac{13}{18}\\pi, \\; \\frac{17}{18}\\pi$$"}]}],"generatedAt":"2026-06-09T05:12:19.153Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"3e87f81a-b53f-40d4-be9c-9634b4b7f78b:trigonometric_identities","label":"(a) Using Trigonometric Identities","steps":[{"type":"text","order":0,"title":"Apply addition formula","content":"To express $\\sin 3\\theta$ in terms of $\\sin \\theta$, we start by using the addition formula (also known as the compound angle identity) for sine:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nApplying this identity to $\\sin(2\\theta + \\theta)$, we get:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos \\theta + \\cos 2\\theta \\sin \\theta$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$${\\sin 3\\theta} = \\htmlData{anchor=sin2t-eq}{\\color{@sky}{\\sin 2\\theta}} \\cos \\theta + \\htmlData{anchor=cos2t-eq}{\\color{@amber}{\\cos 2\\theta}} \\sin \\theta$"},{"latex":"$${\\sin 3\\theta} = \\htmlData{anchor=sin2t-sub}{\\color{@sky}{(2\\sin\\theta\\cos\\theta)}} \\cos\\theta + \\htmlData{anchor=cos2t-sub}{\\color{@amber}{(1 - 2\\sin^2\\theta)}} \\sin\\theta$"},{"text":"Next, we replace the double-angle terms using the standard double-angle formulas:"},{"text":"- $\\sin 2\\theta = 2\\sin\\theta\\cos\\theta$"},{"text":"- $\\cos 2\\theta = 1 - 2\\sin^2\\theta$ (choosing this version to work directly towards terms containing sine)"}],"highlights":[{"color":"sky","style":"underline","anchor":"sin2t-eq"},{"color":"amber","style":"underline","anchor":"cos2t-eq"}],"connections":[{"to":"sin2t-sub","from":"sin2t-eq","color":"sky","label":"Substitute $2\\sin\\theta\\cos\\theta$","style":"solid"},{"to":"cos2t-sub","from":"cos2t-eq","color":"amber","label":"Substitute $1 - 2\\sin^2\\theta$","style":"solid"}]},"order":1,"title":"Substitute double-angle formulas"},{"type":"text","order":2,"title":"Simplify using identity","content":"Now we simplify the expression:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta\\cos^2\\theta + \\sin\\theta(1 - 2\\sin^2\\theta) \\\\ &= 2\\sin\\theta\\cos^2\\theta + \\sin\\theta - 2\\sin^3\\theta \\end{aligned}$$\n\nTo make the entire equation in terms of $\\sin\\theta$, we substitute the Pythagorean identity $\\cos^2\\theta = 1 - \\sin^2\\theta$ into the first term:\n\n$$\\sin 3\\theta = 2\\sin\\theta(1 - \\sin^2\\theta) + \\sin\\theta - 2\\sin^3\\theta$$"},{"type":"text","order":3,"title":"Expand and collect terms","content":"Expand the first term and group all the like terms together:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta - 2\\sin^3\\theta + \\sin\\theta - 2\\sin^3\\theta \\\\ &= (2\\sin\\theta + \\sin\\theta) + (-2\\sin^3\\theta - 2\\sin^3\\theta) \\\\ &= 3\\sin\\theta - 4\\sin^3\\theta \\end{aligned}$$\n\nThis completes the proof for Part 1."},{"type":"text","order":4,"title":"Identify the identity","content":"For Part 2, we need to solve:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.5$$\n\nUsing the identity we proved in Part 1, we can rewrite the left-hand side of the equation as $\\sin 3\\theta$:\n\n$$\\sin 3\\theta = 0.5$$"},{"type":"text","order":5,"title":"Find the general solutions","content":"We are given the interval $0 \\leq \\theta \\leq \\pi$. \n\nMultiplying the inequality by $3$, we find the corresponding range for $3\\theta$:\n\n$$0 \\leq 3\\theta \\leq 3\\pi$$\n\nWithin the interval $[0, 3\\pi]$, we find the values where the sine function is equal to $0.5$:\n\n$$\\begin{aligned} 3\\theta &= \\frac{\\pi}{6}, \\pi - \\frac{\\pi}{6}, 2\\pi + \\frac{\\pi}{6}, 3\\pi - \\frac{\\pi}{6} \\\\ 3\\theta &= \\frac{\\pi}{6}, \\frac{5\\pi}{6}, \\frac{13\\pi}{6}, \\frac{17\\pi}{6} \\end{aligned}$$"},{"type":"text","order":6,"title":"Solve for theta","content":"To find $\\theta$, we divide each value by $3$:\n\n$$\\theta = \\frac{\\pi}{18}, \\frac{5\\pi}{18}, \\frac{13\\pi}{18}, \\frac{17\\pi}{18}$$\n\nThese four solutions all lie within the given range $0 \\leq \\theta \\leq \\pi$."}]},{"id":"7f84acd4-2695-49e2-969b-a71a991e5694:triple_angle_identity","label":"(b) Triple Angle Identity","steps":[{"type":"text","order":0,"title":"Prove the identity","content":"For Part 1, we show that $\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$ by expanding $\\sin(2\\theta + \\theta)$ using the sine compound angle identity:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nLetting $A = 2\\theta$ and $B = \\theta$, we get:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos \\theta + \\cos 2\\theta \\sin \\theta$$\n\nNext, we use the double-angle formulas $\\sin 2\\theta = 2\\sin\\theta\\cos\\theta$ and $\\cos 2\\theta = 1 - 2\\sin^2\\theta$ to substitute:\n\n$$\\sin 3\\theta = (2\\sin\\theta\\cos\\theta)\\cos\\theta + (1 - 2\\sin^2\\theta)\\sin\\theta$$\n\n$$\\sin 3\\theta = 2\\sin\\theta\\cos^2\\theta + \\sin\\theta - 2\\sin^3\\theta$$\n\nUsing the fundamental identity $\\cos^2\\theta = 1 - \\sin^2\\theta$, we express everything in terms of $\\sin\\theta$:\n\n$$\\sin 3\\theta = 2\\sin\\theta(1 - \\sin^2\\theta) + \\sin\\theta - 2\\sin^3\\theta$$\n\n$$\\sin 3\\theta = 2\\sin\\theta - 2\\sin^3\\theta + \\sin\\theta - 2\\sin^3\\theta$$\n\n$$\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$$\n\nThis completes the proof for Part 1."},{"type":"text","order":1,"title":"Recognise triple angle identity","content":"For Part 2, the left-hand side of the given equation is $3\\sin\\theta - 4\\sin^3\\theta$. We recognise this as the triple-angle identity we proved in Part 1:\n\n$$\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$$\n\nSubstituting this identity into our equation, we can rewrite it in a much simpler form:\n\n$$\\sin 3\\theta = 0.5$$"},{"type":"text","order":2,"title":"Find the new interval","content":"We are given the interval for $\\theta$ as:\n\n$$0 \\leq \\theta \\leq \\pi$$\n\nSince our equation is now in terms of $3\\theta$, we must multiply the entire inequality by $3$ to find the corresponding interval for $3\\theta$:\n\n$$0 \\leq 3\\theta \\leq 3\\pi$$"},{"type":"text","order":3,"title":"Solve for 3theta","content":"Now we find all solutions to $\\sin 3\\theta = 0.5$ within the interval $[0, 3\\pi]$.\n\nThe principal value is:\n\n$$3\\theta = \\arcsin(0.5) = \\frac{\\pi}{6}$$\n\nUsing the symmetry of the sine graph, the second solution is:\n\n$$3\\theta = \\pi - \\frac{\\pi}{6} = \\frac{5\\pi}{6}$$\n\nTo find the rest of the solutions within the range $[0, 3\\pi]$, we add the period $2\\pi$ to each of our first two solutions:\n\n$$3\\theta = \\frac{\\pi}{6} + 2\\pi = \\frac{13\\pi}{6}$$\n\n$$3\\theta = \\frac{5\\pi}{6} + 2\\pi = \\frac{17\\pi}{6}$$\n\nThese four values lie within $[0, 3\\pi]$:\n\n$$3\\theta = \\frac{\\pi}{6}, \\; \\frac{5\\pi}{6}, \\; \\frac{13\\pi}{6}, \\; \\frac{17\\pi}{6}$$"},{"type":"text","order":4,"title":"Solve for theta","content":"To find the values of $\\theta$, we divide each of our solutions for $3\\theta$ by $3$:\n\n$$\\theta = \\frac{\\pi}{18}, \\; \\frac{5\\pi}{18}, \\; \\frac{13\\pi}{18}, \\; \\frac{17\\pi}{18}$$\n\nEach of these values lies within the original interval $[0, \\pi]$, so they are all valid solutions."}]},{"id":"7f84acd4-2695-49e2-969b-a71a991e5694:graphing_calculator","label":"(b) Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the functions","content":"To solve the equation using a graphing calculator, we first set up two functions to graph. We will look for their intersection points within the given interval:\n\n$$y_1 = 3\\sin\\theta - 4\\sin^3\\theta$$\n$$y_2 = 0.5$$\n\nWe restrict our view to the domain $0 \\leq \\theta \\leq \\pi$. On your calculator, use the variable $x$ instead of $\\theta$."},{"type":"text","order":1,"title":"Graph and find intersections","content":"Set your calculator's angle mode to **Radians**. \n\nEnter $y_1 = 3\\sin(x) - 4(\\sin(x))^3$ and $y_2 = 0.5$, then adjust your window settings so you can clearly see the intersections from $x = 0$ to $x = \\pi$ (approximately $0$ to $3.1416$). \n\nUse the intersection tool to find the $x$-coordinates of the four points where the curves cross.","calculator":[{"gdc":{"expression":"y = 3*sin(x) - 4*(sin(x))^3","instructions":"1. Go to Graph mode and set your angle unit to Radians.\n2. Enter the functions:\n $f1(x) = 3\\sin(x) - 4(\\sin(x))^3$\n $f2(x) = 0.5$\n3. Adjust the view window so that $x$ ranges from 0 to $\\pi$.\n4. Use the G-Solv or Analyze Graph tool to find the intersections."},"ti84":{"expression":"y = 3*sin(x) - 4*(sin(x))^3","instructions":"1. Press [MODE], ensure 'RADIAN' is highlighted.\n2. Press [Y=] and enter:\n $Y_1 = 3\\sin(X) - 4(\\sin(X))^3$\n $Y_2 = 0.5$\n3. Press [WINDOW] and set: \n $X_{min} = 0$, $X_{max} = \\pi$, $Y_{min} = -1.5$, $Y_{max} = 1.5$.\n4. Press [GRAPH].\n5. Press [2nd] [CALC] and select '5: intersect'. Select the first curve, second curve, and guess near each intersection point to find all four coordinates."},"desmos":{"expression":"y = 3*sin(x) - 4*(\\sin(x))^3 \\{0 \\le x \\le \\pi\\}","instructions":"1. Ensure the graph is in Radians mode (click the wrench icon in the top right).\n2. Type the equations:\n $y = 3\\sin(x) - 4(\\sin(x))^3$\n $y = 0.5$\n3. Restrict the domain of the first function by typing $\\{0 \\le x \\le \\pi\\}$.\n4. Click on the gray intersection points to reveal their coordinates."},"description":"Graph the functions and find the intersection points in Radians mode."}]},{"type":"text","order":2,"title":"Record the decimal values","content":"From the graph, we find four intersection points in the interval $[0, \\pi]$. The approximate $x$-values (representing $\\theta$) are:\n\n* $\\theta_1 \\approx 0.17453$\n* $\\theta_2 \\approx 0.87266$\n* $\\theta_3 \\approx 2.26893$\n* $\\theta_4 \\approx 2.96706$"},{"type":"text","order":3,"title":"Convert to exact multiples","content":"To write these decimal values in terms of $\\pi$ as requested by the question, divide each value by $\\pi$ to find the corresponding fractional coefficient:\n\n$$\\frac{0.17453}{\\pi} \\approx 0.05556 \\approx \\frac{1}{18} \\implies \\theta_1 = \\frac{1}{18}\\pi$$\n\n$$\\frac{0.87266}{\\pi} \\approx 0.27778 \\approx \\frac{5}{18} \\implies \\theta_2 = \\frac{5}{18}\\pi$$\n\n$$\\frac{2.26893}{\\pi} \\approx 0.72222 \\approx \\frac{13}{18} \\implies \\theta_3 = \\frac{13}{18}\\pi$$\n\n$$\\frac{2.96706}{\\pi} \\approx 0.94444 \\approx \\frac{17}{18} \\implies \\theta_4 = \\frac{17}{18}\\pi$$\n\nThus, the four solutions in the interval $0 \\leq \\theta \\leq \\pi$ are:\n\n$$\\theta = \\frac{1}{18}\\pi, \\; \\frac{5}{18}\\pi, \\; \\frac{13}{18}\\pi, \\; \\frac{17}{18}\\pi$$"}]}],"generatedAt":"2026-06-09T05:12:27.300Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"938f7bbf-f10d-4cd5-a0b0-f8e7ed8cfd61","revisionId":"e156f51a-8f32-4c52-9c85-8264e6ca2580","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"d1def23c-4d8d-4cc7-9a88-b8686ba79f88","label":"a","content":"Solve, for $0 \\leq x \\leq 360$, the equation,\n\n$$6 \\sin x \\cos x = 1.8$$\n\nGive your answers to 1 decimal place.","markscheme":"- Uses $2 \\sin x \\cos x = \\sin 2x$ to rewrite as $3 \\sin 2x = 1.8$ M1\n- $\\sin 2x = 0.6$ M1\n- Finds values for $2x$ in the range $[0, 720]$: $36.87, 143.13, 396.87, 503.13$ M1\n- Divides by 2 to find $x$ M1\n- Final answers: $18.4, 71.6, 198.4, 251.6$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Uses $2 \\sin x \\cos x = \\sin 2x$ to rewrite the equation.","code":"M"},{"id":"p2","marks":1,"description":"Obtains $\\sin 2x = 0.6$.","code":"M"},{"id":"p3","marks":1,"description":"Finds at least two correct values for $2x$ (e.g., $36.87, 143.13$).","code":"M"},{"id":"p4","marks":1,"description":"Divides the values of $2x$ by 2 to find $x$.","code":"M"},{"id":"p5","marks":1,"description":"Provides all four correct final answers to 1 decimal place: $18.4, 71.6, 198.4, 251.6$.","code":"A"}],"finalAnswers":["18.4, 71.6, 198.4, 251.6"],"generatedAt":"2026-05-06T14:03:54.977Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"double_angle_identity","label":"Double Angle Identity","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$6 \\sin x \\cos x = 1.8$"},{"latex":"$$3 \\cdot (\\htmlData{anchor=double-angle-identity}{\\color{@sky}{2 \\sin x \\cos x}}) = 1.8$"},{"latex":"$$3 \\htmlData{anchor=double-angle-sub}{\\color{@sky}{\\sin 2x}} = 1.8$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"double-angle-identity"},{"color":"sky","style":"text-color","anchor":"double-angle-sub"}],"connections":[{"to":"double-angle-sub","from":"double-angle-identity","color":"sky","label":"identity","style":"solid"}]},"order":0,"title":"Apply double angle identity","description":"We can rewrite the equation by recalling the double angle identity for sine:\n\n$$2 \\sin x \\cos x = \\sin 2x$$\n\nSplitting $6 \\sin x \\cos x$ into $3 \\times (2 \\sin x \\cos x)$ allows us to substitute this identity to simplify the expression."},{"type":"text","order":1,"title":"Solve for sin 2x","content":"Now, divide both sides of the equation by $3$ to isolate the trigonometric term:\n\n$$\\begin{aligned} 3 \\sin 2x &= 1.8 \\\\ \\sin 2x &= \\frac{1.8}{3} \\\\ \\sin 2x &= 0.6 \\end{aligned}$$"},{"type":"text","order":2,"title":"Adjust the domain","content":"Our original range is given as $0^\\circ \\le x \\le 360^\\circ$.\n\nSince we are solving for $2x$, we must adjust our search interval by multiplying the entire inequality by $2$:\n\n$$0^\\circ \\le 2x \\le 720^\\circ$$"},{"type":"text","order":3,"title":"Find all values of 2x","content":"First, find the principal value using the inverse sine function:\n\n$$2x = \\sin^{-1}(0.6) \\approx 36.87^\\circ$$\n\nNext, use the symmetry properties of the sine function to find all other solutions within the expanded range $[0^\\circ, 720^\\circ]$:\n\n* **First quadrant (principal angle):** $2x_1 = 36.87^\\circ$\n* **Second quadrant ($180^\\circ - \\theta$):** $2x_2 = 180^\\circ - 36.87^\\circ = 143.13^\\circ$\n* **Second cycle ($+360^\\circ$):**\n $$2x_3 = 36.87^\\circ + 360^\\circ = 396.87^\\circ$$\n $$2x_4 = 143.13^\\circ + 360^\\circ = 503.13^\\circ$$","calculator":[{"gdc":{"expression":"sin⁻¹(0.6)","instructions":"1. Set the calculator angle mode to degrees in the setup menu.\n2. In the run-matrix menu, input $\\sin^{-1}(0.6)$ and press [EXE]."},"ti84":{"expression":"sin⁻¹(0.6)","instructions":"1. Press [MODE] and ensure 'DEGREE' is highlighted, then press [ENTER].\n2. Press [2nd] then [SIN] to select the $\\sin^{-1}$ function.\n3. Type '0.6' and press [ENTER]."},"desmos":{"expression":"\\arcsin(0.6)","instructions":"1. Click the wrench icon in the top-right corner to open Settings and select 'Degrees'.\n2. Type 'arcsin(0.6)' or 'sin^-1(0.6)' into the input bar."},"description":"Calculate the principal angle $\\sin^{-1}(0.6)$ in degrees."}]},{"type":"text","order":4,"title":"Divide by 2","content":"Divide each angle value by $2$ to obtain the values for $x$:\n\n$$\\begin{aligned} x_1 &= \\frac{36.87^\\circ}{2} = 18.435^\\circ \\approx 18.4^\\circ \\\\ x_2 &= \\frac{143.13^\\circ}{2} = 71.565^\\circ \\approx 71.6^\\circ \\\\ x_3 &= \\frac{396.87^\\circ}{2} = 198.435^\\circ \\approx 198.4^\\circ \\\\ x_4 &= \\frac{503.13^\\circ}{2} = 251.565^\\circ \\approx 251.6^\\circ \\end{aligned}$$\n\nAll of these values lie within the original domain of $0^\\circ \\le x \\le 360^\\circ$. Rounded to 1 decimal place, the final set of solutions is:\n\n$$x = 18.4^\\circ, \\; 71.6^\\circ, \\; 198.4^\\circ, \\; 251.6^\\circ$$"}]},{"id":"graphical_calculator","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the functions","content":"To solve the trigonometric equation using a graphing calculator, we can find the intersection of two separate functions representing the left-hand side and right-hand side of the equation:\n\n* Let $Y_1 = 6\\sin x\\cos x$\n* Let $Y_2 = 1.8$\n\nWe need to find all points of intersection in the interval $[0^\\circ, 360^\\circ]$. Make sure your calculator is set to **Degree** mode."},{"type":"text","order":1,"title":"Configure calculator settings","content":"Set the viewing window of your graphing calculator to display the entire domain of $x$ from $0^\\circ$ to $360^\\circ$ and the $y$-values around $1.8$. Appropriate window settings are:\n* $X_{\\text{min}} = 0$\n* $X_{\\text{max}} = 360$\n* $Y_{\\text{min}} = -4$\n* $Y_{\\text{max}} = 4$","calculator":[{"gdc":{"expression":"f1(x) = 6*sin(x)*cos(x), f2(x) = 1.8","instructions":"1. Change the angle setting to Degrees in the system/setup menu.\n2. In the Graph application, enter \$f1(x) = 6\\sin(x)\\cos(x)\$ and \$f2(x) = 1.8\$.\n3. Set the window dimensions with \$x\$ from 0 to 360 and \$y\$ from -4 to 4.\n4. Use the 'Analyze Graph' -> 'Intersection' tool. Drag the shaded region across each intersection to read the coordinate values."},"ti84":{"expression":"Y1 = 6*sin(X)*cos(X), Y2 = 1.8","instructions":"1. Press [MODE], ensure DEGREE is highlighted, and press [ENTER].\n2. Press [Y=], enter \$Y_1 = 6\\sin(X)\\cos(X)\$ and \$Y_2 = 1.8\$.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Xscl = 45, Ymin = -4, Ymax = 4.\n4. Press [GRAPH] to display the curves.\n5. Press [2nd] [TRACE] (CALC) and select '5: intersect'.\n6. Use the arrow keys to move the cursor close to an intersection, then press [ENTER] three times. Repeat for all 4 intersection points."},"desmos":{"expression":"y = 6\\sin(x)\\cos(x)\ny = 1.8","instructions":"1. Click the wrench icon (Graph Settings) in the top-right corner.\n2. Select 'Degrees'. Set the X-axis limits to [0, 360] and Y-axis limits to [-4, 4].\n3. Type `y = 6\\sin(x)\\cos(x)` into the first input bar.\n4. Type `y = 1.8` into the second input bar.\n5. Click on the gray intersection points of the two curves to display their coordinates."},"description":"Graph the functions and find their intersection points on your calculator."}]},{"type":"text","order":2,"title":"Find the intersection points","content":"By tracing the intersections of the two plotted curves on your calculator within the interval $[0^\\circ, 360^\\circ]$, we locate exactly four points of intersection:\n\n* First intersection: $x \\approx 18.435^\\circ$\n* Second intersection: $x \\approx 71.565^\\circ$\n* Third intersection: $x \\approx 198.435^\\circ$\n* Fourth intersection: $x \\approx 251.565^\\circ$"},{"type":"text","order":3,"title":"Round to one decimal place","content":"Rounding each of the obtained values to $1$ decimal place as requested gives the final solutions:\n\n* $x = 18.4^\\circ$\n* $x = 71.6^\\circ$\n* $x = 198.4^\\circ$\n* $x = 251.6^\\circ$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]}],"generatedAt":"2026-06-09T07:02:13.005Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"d1def23c-4d8d-4cc7-9a88-b8686ba79f88:double_angle_identity","label":"Double Angle Identity","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$6 \\sin x \\cos x = 1.8$"},{"latex":"$$3 \\cdot (\\htmlData{anchor=double-angle-identity}{\\color{@sky}{2 \\sin x \\cos x}}) = 1.8$"},{"latex":"$$3 \\htmlData{anchor=double-angle-sub}{\\color{@sky}{\\sin 2x}} = 1.8$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"double-angle-identity"},{"color":"sky","style":"text-color","anchor":"double-angle-sub"}],"connections":[{"to":"double-angle-sub","from":"double-angle-identity","color":"sky","label":"identity","style":"solid"}]},"order":0,"title":"Apply double angle identity","description":"We can rewrite the equation by recalling the double angle identity for sine:\n\n$$2 \\sin x \\cos x = \\sin 2x$$\n\nSplitting $6 \\sin x \\cos x$ into $3 \\times (2 \\sin x \\cos x)$ allows us to substitute this identity to simplify the expression."},{"type":"text","order":1,"title":"Solve for sin 2x","content":"Now, divide both sides of the equation by $3$ to isolate the trigonometric term:\n\n$$\\begin{aligned} 3 \\sin 2x &= 1.8 \\\\ \\sin 2x &= \\frac{1.8}{3} \\\\ \\sin 2x &= 0.6 \\end{aligned}$$"},{"type":"text","order":2,"title":"Adjust the domain","content":"Our original range is given as $0^\\circ \\le x \\le 360^\\circ$.\n\nSince we are solving for $2x$, we must adjust our search interval by multiplying the entire inequality by $2$:\n\n$$0^\\circ \\le 2x \\le 720^\\circ$$"},{"type":"text","order":3,"title":"Find all values of 2x","content":"First, find the principal value using the inverse sine function:\n\n$$2x = \\sin^{-1}(0.6) \\approx 36.87^\\circ$$\n\nNext, use the symmetry properties of the sine function to find all other solutions within the expanded range $[0^\\circ, 720^\\circ]$:\n\n* **First quadrant (principal angle):** $2x_1 = 36.87^\\circ$\n* **Second quadrant ($180^\\circ - \\theta$):** $2x_2 = 180^\\circ - 36.87^\\circ = 143.13^\\circ$\n* **Second cycle ($+360^\\circ$):**\n $$2x_3 = 36.87^\\circ + 360^\\circ = 396.87^\\circ$$\n $$2x_4 = 143.13^\\circ + 360^\\circ = 503.13^\\circ$$","calculator":[{"gdc":{"expression":"sin⁻¹(0.6)","instructions":"1. Set the calculator angle mode to degrees in the setup menu.\n2. In the run-matrix menu, input $\\sin^{-1}(0.6)$ and press [EXE]."},"ti84":{"expression":"sin⁻¹(0.6)","instructions":"1. Press [MODE] and ensure 'DEGREE' is highlighted, then press [ENTER].\n2. Press [2nd] then [SIN] to select the $\\sin^{-1}$ function.\n3. Type '0.6' and press [ENTER]."},"desmos":{"expression":"\\arcsin(0.6)","instructions":"1. Click the wrench icon in the top-right corner to open Settings and select 'Degrees'.\n2. Type 'arcsin(0.6)' or 'sin^-1(0.6)' into the input bar."},"description":"Calculate the principal angle $\\sin^{-1}(0.6)$ in degrees."}]},{"type":"text","order":4,"title":"Divide by 2","content":"Divide each angle value by $2$ to obtain the values for $x$:\n\n$$\\begin{aligned} x_1 &= \\frac{36.87^\\circ}{2} = 18.435^\\circ \\approx 18.4^\\circ \\\\ x_2 &= \\frac{143.13^\\circ}{2} = 71.565^\\circ \\approx 71.6^\\circ \\\\ x_3 &= \\frac{396.87^\\circ}{2} = 198.435^\\circ \\approx 198.4^\\circ \\\\ x_4 &= \\frac{503.13^\\circ}{2} = 251.565^\\circ \\approx 251.6^\\circ \\end{aligned}$$\n\nAll of these values lie within the original domain of $0^\\circ \\le x \\le 360^\\circ$. Rounded to 1 decimal place, the final set of solutions is:\n\n$$x = 18.4^\\circ, \\; 71.6^\\circ, \\; 198.4^\\circ, \\; 251.6^\\circ$$"}]},{"id":"d1def23c-4d8d-4cc7-9a88-b8686ba79f88:graphical_calculator","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the functions","content":"To solve the trigonometric equation using a graphing calculator, we can find the intersection of two separate functions representing the left-hand side and right-hand side of the equation:\n\n* Let $Y_1 = 6\\sin x\\cos x$\n* Let $Y_2 = 1.8$\n\nWe need to find all points of intersection in the interval $[0^\\circ, 360^\\circ]$. Make sure your calculator is set to **Degree** mode."},{"type":"text","order":1,"title":"Configure calculator settings","content":"Set the viewing window of your graphing calculator to display the entire domain of $x$ from $0^\\circ$ to $360^\\circ$ and the $y$-values around $1.8$. Appropriate window settings are:\n* $X_{\\text{min}} = 0$\n* $X_{\\text{max}} = 360$\n* $Y_{\\text{min}} = -4$\n* $Y_{\\text{max}} = 4$","calculator":[{"gdc":{"expression":"f1(x) = 6*sin(x)*cos(x), f2(x) = 1.8","instructions":"1. Change the angle setting to Degrees in the system/setup menu.\n2. In the Graph application, enter \$f1(x) = 6\\sin(x)\\cos(x)\$ and \$f2(x) = 1.8\$.\n3. Set the window dimensions with \$x\$ from 0 to 360 and \$y\$ from -4 to 4.\n4. Use the 'Analyze Graph' -> 'Intersection' tool. Drag the shaded region across each intersection to read the coordinate values."},"ti84":{"expression":"Y1 = 6*sin(X)*cos(X), Y2 = 1.8","instructions":"1. Press [MODE], ensure DEGREE is highlighted, and press [ENTER].\n2. Press [Y=], enter \$Y_1 = 6\\sin(X)\\cos(X)\$ and \$Y_2 = 1.8\$.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Xscl = 45, Ymin = -4, Ymax = 4.\n4. Press [GRAPH] to display the curves.\n5. Press [2nd] [TRACE] (CALC) and select '5: intersect'.\n6. Use the arrow keys to move the cursor close to an intersection, then press [ENTER] three times. Repeat for all 4 intersection points."},"desmos":{"expression":"y = 6\\sin(x)\\cos(x)\ny = 1.8","instructions":"1. Click the wrench icon (Graph Settings) in the top-right corner.\n2. Select 'Degrees'. Set the X-axis limits to [0, 360] and Y-axis limits to [-4, 4].\n3. Type `y = 6\\sin(x)\\cos(x)` into the first input bar.\n4. Type `y = 1.8` into the second input bar.\n5. Click on the gray intersection points of the two curves to display their coordinates."},"description":"Graph the functions and find their intersection points on your calculator."}]},{"type":"text","order":2,"title":"Find the intersection points","content":"By tracing the intersections of the two plotted curves on your calculator within the interval $[0^\\circ, 360^\\circ]$, we locate exactly four points of intersection:\n\n* First intersection: $x \\approx 18.435^\\circ$\n* Second intersection: $x \\approx 71.565^\\circ$\n* Third intersection: $x \\approx 198.435^\\circ$\n* Fourth intersection: $x \\approx 251.565^\\circ$"},{"type":"text","order":3,"title":"Round to one decimal place","content":"Rounding each of the obtained values to $1$ decimal place as requested gives the final solutions:\n\n* $x = 18.4^\\circ$\n* $x = 71.6^\\circ$\n* $x = 198.4^\\circ$\n* $x = 251.6^\\circ$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]}],"generatedAt":"2026-06-09T07:02:27.119Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"04df7d63-f5cb-4485-9727-ce5baaaf8bdc","revisionId":"e945c6eb-bc78-4540-9b54-dd65dccb8d49","questionNumber":"2","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"7.1 Addition Formulae","calculatorUse":null,"isCalculation":true,"parts":[{"id":"1ba3f6ca-faaa-4fd7-ac2a-9a95f1605f51","label":"a","content":"By writing $\\sin 3\\theta$ as $\\sin(2\\theta + \\theta)$ show that $\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$","markscheme":"- Expands $\\sin(2\\theta + \\theta)$ to $\\sin 2\\theta \\cos \\theta + \\cos 2\\theta \\sin \\theta$ M1\n- Substitutes $2 \\sin \\theta \\cos \\theta$ for $\\sin 2\\theta$ and $(1 - 2 \\sin^2 \\theta)$ for $\\cos 2\\theta$ and simplifies to the required form A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Applies the addition formula for $\\sin(2\\theta + \\theta)$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly substitutes double angle formulae and simplifies to $3\\sin\\theta - 4\\sin^3\\theta$.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T03:35:10.313Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_sine_substitution","label":"Direct Sine Substitution","steps":[{"type":"text","order":0,"title":"Use addition formula","content":"To show the identity for $\\sin 3\\theta$, we begin by using the addition formula for sine:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nBy writing $3\\theta$ as $2\\theta + \\theta$, we let $A = 2\\theta$ and $B = \\theta$ to expand the expression:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos \\theta + \\cos 2\\theta \\sin \\theta$$\n\nThis earns the first mark M1 for a correct compound angle expansion."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sin 2\\theta = \\htmlData{anchor=sin2-sub}{\\color{@sky}{2\\sin\\theta\\cos\\theta}}$"},{"latex":"$$\\cos 2\\theta = \\htmlData{anchor=cos2-sub}{\\color{@amber}{1 - 2\\sin^2\\theta}}$"},{"latex":"$$\\sin(2\\theta + \\theta) = \\left(\\htmlData{anchor=sin2-term}{\\color{@sky}{2\\sin\\theta\\cos\\theta}}\\right)\\cos\\theta + \\left(\\htmlData{anchor=cos2-term}{\\color{@amber}{1 - 2\\sin^2\\theta}}\\right)\\sin\\theta$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sin2-sub"},{"color":"amber","style":"text-color","anchor":"cos2-sub"}],"connections":[{"to":"sin2-term","from":"sin2-sub","color":"sky","label":"substitute","style":"solid"},{"to":"cos2-term","from":"cos2-sub","color":"amber","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute double-angle formulae","description":"Next, we substitute the double-angle identities. Following the Direct Sine Substitution method, we replace $\\sin 2\\theta$ with $2\\sin\\theta\\cos\\theta$ and $\\cos 2\\theta$ directly with $1 - 2\\sin^2\\theta$."},{"type":"text","order":2,"title":"Convert remaining cosines","content":"Let's simplify the expression we obtained:\n\n$$\\sin 3\\theta = 2\\sin\\theta\\cos^2\\theta + \\sin\\theta - 2\\sin^3\\theta$$\n\nTo write the entire equation in terms of $\\sin\\theta$, we convert the remaining $\\cos^2\\theta$ term using the fundamental identity $\\cos^2\\theta = 1 - \\sin^2\\theta$:\n\n$$\\sin 3\\theta = 2\\sin\\theta(1 - \\sin^2\\theta) + \\sin\\theta - 2\\sin^3\\theta$$"},{"type":"text","order":3,"title":"Expand and simplify","content":"Now, we expand the brackets and collect the like terms to arrive at our final result:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta - 2\\sin^3\\theta + \\sin\\theta - 2\\sin^3\\theta \\\\ &= 3\\sin\\theta - 4\\sin^3\\theta \\end{aligned}$$\n\nThis completes the proof of the identity and earns the final mark A1."},{"type":"text","order":4,"title":"Recognise the equation form","content":"For Part 2, we are given the equation:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\nRecognising that the left-hand side is exactly the identity we just proved for $\\sin 3\\theta$, we can rewrite the equation as:\n\n$$\\sin 3\\theta = 0.4$$\n\nThis step earns the first mark M1."},{"type":"text","order":5,"title":"Find intervals and solve","content":"Since $0^\\circ \\leq \\theta \\leq 180^\\circ$, the interval for $3\\theta$ is:\n\n$$0^\\circ \\leq 3\\theta \\leq 540^\\circ$$\n\nWe find the principal value and secondary values in this interval:\n\n* **First quadrant:** $3\\theta \\approx 23.578^\\circ$\n* **Second quadrant:** $3\\theta = 180^\\circ - 23.578^\\circ \\approx 156.422^\\circ$\n* **Next cycle (add $360^\\circ$):**\n * $3\\theta = 23.578^\\circ + 360^\\circ \\approx 383.578^\\circ$\n * $3\\theta = 156.422^\\circ + 360^\\circ \\approx 516.422^\\circ$\n\nFinding these values for $3\\theta$ earns another mark M1."},{"type":"text","order":6,"title":"Divide to find theta","content":"Now, we divide each value of $3\\theta$ by $3$ to solve for $\\theta$:\n\n* $\\theta = \\frac{23.578^\\circ}{3} \\approx 7.9^\\circ$\n* $\\theta = \\frac{156.422^\\circ}{3} \\approx 52.1^\\circ$\n* $\\theta = \\frac{383.578^\\circ}{3} \\approx 127.9^\\circ$\n* $\\theta = \\frac{516.422^\\circ}{3} \\approx 172.1^\\circ$\n\nDividing by 3 earns M1, and giving the final answers to 1 decimal place earns A1:\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$"}]},{"id":"general_cosine_substitution","label":"General Cosine Substitution","steps":[{"type":"text","order":0,"title":"Apply compound angle formula","content":"We begin by writing $\\sin 3\\theta$ as $\\sin(2\\theta + \\theta)$ and using the compound angle formula for sine:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nApplying this with $A = 2\\theta$ and $B = \\theta$ gives:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos\\theta + \\cos 2\\theta \\sin\\theta$$\n\nThis key step earns the first method mark M1 on the markscheme."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Identities: } \\htmlData{anchor=sin2-id}{\\color{@sky}{\\sin 2\\theta = 2\\sin\\theta\\cos\\theta}}, \\quad \\htmlData{anchor=cos2-id}{\\color{@amber}{\\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta}}$"},{"latex":"$$\\sin 3\\theta = \\htmlData{anchor=sin2-sub}{\\color{@sky}{\\sin 2\\theta}} \\cos\\theta + \\htmlData{anchor=cos2-sub}{\\color{@amber}{\\cos 2\\theta}} \\sin\\theta$"},{"latex":"$$\\phantom{\\sin 3\\theta} = (\\color{@sky}{2\\sin\\theta\\cos\\theta})\\cos\\theta + (\\color{@amber}{\\cos^2\\theta - \\sin^2\\theta})\\sin\\theta$"}],"highlights":[{"color":"sky","style":"box","anchor":"sin2-sub"},{"color":"amber","style":"box","anchor":"cos2-sub"}],"connections":[{"to":"sin2-sub","from":"sin2-id","color":"sky","label":"substitute","style":"solid"},{"to":"cos2-sub","from":"cos2-id","color":"amber","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute double-angle formulae","description":"We now substitute the double-angle formulae. For $\\sin 2\\theta$, we use the standard identity. For $\\cos 2\\theta$, we use the general cosine double-angle identity $\\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta$."},{"type":"text","order":2,"title":"Group the terms","content":"Now we expand the brackets and group the terms containing $\\cos^2\\theta$:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta\\cos^2\\theta + \\sin\\theta\\cos^2\\theta - \\sin^3\\theta \\\\ &= 3\\sin\\theta\\cos^2\\theta - \\sin^3\\theta \\end{aligned}$$"},{"type":"text","order":3,"title":"Convert cosine to sine","content":"To express everything solely in terms of $\\sin\\theta$, we use the Pythagorean identity $\\cos^2\\theta = 1 - \\sin^2\\theta$ to substitute for the remaining $\\cos^2\\theta$ term:\n\n$$\\begin{aligned} \\sin 3\\theta &= 3\\sin\\theta(1 - \\sin^2\\theta) - \\sin^3\\theta \\\\ &= 3\\sin\\theta - 3\\sin^3\\theta - \\sin^3\\theta \\\\ &= 3\\sin\\theta - 4\\sin^3\\theta \\end{aligned}$$\n\nThis completes the proof and earns the final accuracy mark A1."},{"type":"text","order":4,"title":"Recognise the equation form","content":"For Part 2, we must solve the equation:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\nSubstituting the identity $\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$ simplifies this equation to:\n\n$$\\sin 3\\theta = 0.4$$\n\nThis yields the first method mark M1."},{"type":"text","order":5,"title":"Find intervals and solve","content":"Given the boundary condition $0^\\circ \\leq \\theta \\leq 180^\\circ$, the interval for $3\\theta$ expands to:\n\n$$0^\\circ \\leq 3\\theta \\leq 540^\\circ$$\n\nWe determine the valid values of $3\\theta$ within this boundary:\n\n* **First quadrant:** $3\\theta \\approx 23.578^\\circ$\n* **Second quadrant:** $3\\theta = 180^\\circ - 23.578^\\circ \\approx 156.422^\\circ$\n* **Second period (add $360^\\circ$):**\n * $3\\theta \\approx 23.578^\\circ + 360^\\circ = 383.578^\\circ$\n * $3\\theta \\approx 156.422^\\circ + 360^\\circ = 516.422^\\circ$\n\nCorrectly identifying these angles earns the next method mark M1."},{"type":"text","order":6,"title":"Divide to find theta","content":"Finally, we divide each angle by $3$ to find the values for $\\theta$:\n\n* $\\theta = \\frac{23.578^\\circ}{3} \\approx 7.9^\\circ$\n* $\\theta = \\frac{156.422^\\circ}{3} \\approx 52.1^\\circ$\n* $\\theta = \\frac{383.578^\\circ}{3} \\approx 127.9^\\circ$\n* $\\theta = \\frac{516.422^\\circ}{3} \\approx 172.1^\\circ$\n\nDividing by 3 earns M1, and writing the solutions to 1 decimal place earns the final accuracy mark A1:\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$"}]}],"generatedAt":"2026-06-09T03:05:10.812Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"6b66702a-1a9d-459e-b0b5-e98a379d4cd5","label":"b","content":"Solve, for $0 \\leq \\theta \\leq 180$, the equation,\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\nGive your answers to 1 decimal place.","markscheme":"- Recognises the LHS as $\\sin 3\\theta$, so $\\sin 3\\theta = 0.4$ M1\n- Finds values for $3\\theta$: $23.578, 156.422, 383.578, 516.422$ M1\n- Divides by 3 to find $\\theta$ values M1\n- Final answers: $7.9, 52.1, 127.9, 172.1$ A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Identifies the equation as $\\sin 3\\theta = 0.4$.","code":"M"},{"id":"p2","marks":1,"description":"Finds at least two correct values for $3\\theta$ in the expanded range.","code":"M"},{"id":"p3","marks":1,"description":"Divides $3\\theta$ values by 3 to find $\\theta$.","code":"M"},{"id":"p4","marks":1,"description":"Provides all four correct answers: $7.9, 52.1, 127.9, 172.1$.","code":"A"}],"finalAnswers":["7.9","52.1","127.9","172.1"],"generatedAt":"2026-05-06T03:35:10.313Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"triple_angle_identity","label":"Triple Angle Identity","steps":[{"type":"text","order":0,"title":"Apply triple angle identity","content":"Recall the triple-angle identity for sine:\n\n$$\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$$\n\nWe can see that the left-hand side of our equation is exactly the expansion of this identity. Therefore, we can rewrite the equation:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\nas:\n\n$$\\sin 3\\theta = 0.4$$\n\nThis earns the first method mark (**M1**).","highlights":[{"text":"3\\sin\\theta - 4\\sin^3\\theta = 0.4","location":"specification"}]},{"type":"text","order":1,"title":"Determine the interval","content":"Our original interval for $\\theta$ is:\n\n$$0^\\circ \\le \\theta \\le 180^\\circ$$\n\nSince we are solving for $3\\theta$, we must adjust the interval by multiplying all parts by $3$:\n\n$$0^\\circ \\le 3\\theta \\le 540^\\circ$$","highlights":[{"text":"0 \\leq \\theta \\leq 180","location":"specification"}]},{"type":"text","order":2,"title":"Find values for 3θ","content":"Next, we find all possible values of $3\\theta$ within our adjusted interval $[0^\\circ, 540^\\circ]$.\n\n1. Find the principal value using the inverse sine function:\n $$3\\theta_1 = \\arcsin(0.4) \\approx 23.578^\\circ$$\n\n2. Use the symmetry of the sine graph ($\\sin(180^\\circ - x) = \\sin x$) to find the second solution:\n $$3\\theta_2 = 180^\\circ - 23.578^\\circ = 156.422^\\circ$$\n\n3. Add $360^\\circ$ (the period of sine) to find subsequent solutions within $[0^\\circ, 540^\\circ]$:\n $$3\\theta_3 = 23.578^\\circ + 360^\\circ = 383.578^\\circ$$\n $$3\\theta_4 = 156.422^\\circ + 360^\\circ = 516.422^\\circ$$\n\nThese four values for $3\\theta$ earn the second method mark (**M1**)."},{"type":"text","order":3,"title":"Solve for θ","content":"Now, divide each of these values by $3$ to find the corresponding values for $\\theta$ (**M1**):\n\n$$\\begin{aligned} \\theta_1 &= \\frac{23.578^\\circ}{3} \\approx 7.859^\\circ \\\\ \\theta_2 &= \\frac{156.422^\\circ}{3} \\approx 52.141^\\circ \\\\ \\theta_3 &= \\frac{383.578^\\circ}{3} \\approx 127.859^\\circ \\\\ \\theta_4 &= \\frac{516.422^\\circ}{3} \\approx 172.141^\\circ \\end{aligned}$$\n\nRounding each final answer to $1$ decimal place gives (**A1**):\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]},{"id":"cubic_substitution","label":"Substitution and Cubic Equation Solver","steps":[{"type":"text","order":0,"title":"Substitute $x = \\sin\\theta$","content":"To solve this equation using a polynomial approach, we can simplify it by introducing a substitution.\n\nLet $x = \\sin\\theta$. Substituting this into the equation:\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\ngives us:\n$$3x - 4x^3 = 0.4$$\n\nRearranging this into the standard cubic form $ax^3 + bx^2 + cx + d = 0$ by moving all terms to one side, we get:\n$$4x^3 - 3x + 0.4 = 0$$","highlights":[{"text":"3\\sin\\theta - 4\\sin^3\\theta = 0.4","location":"specification"}]},{"type":"text","order":1,"title":"Solve the cubic equation","content":"Using a graphing calculator's polynomial solver, we can find the real roots of this cubic equation:\n$$4x^3 - 3x + 0.4 = 0$$\n\nThe roots (to 4 decimal places) are:\n$$\\begin{aligned}\nx_1 &\\approx 0.1367 \\\\\nx_2 &\\approx 0.7895 \\\\\nx_3 &\\approx -0.9262\n\\end{aligned}$$\n\nSince $x = \\sin\\theta$, we will solve $\\sin\\theta = x$ for each of these three values within the interval $0 \\leq \\theta \\leq 180^{\\circ}$.","calculator":[{"gdc":{"expression":"4x^3 - 3x + 0.4 = 0","instructions":"Go to the Equation/Solver menu. Select 'Polynomial' (F2), and set the degree to 3. Input the coefficients: a = 4, b = 0, c = -3, d = 0.4. Solve to obtain the roots."},"ti84":{"expression":"4x^3 - 3x + 0.4 = 0","instructions":"Press [APPS], select 'PlySmlt2' (Polynomial Root Finder and Simultaneous Equation Solver). Choose 'POLY ROOT FINDER'. Set Order to 3. Enter the coefficients: a3 = 4, a2 = 0, a1 = -3, a0 = 0.4. Press [GRAPH] (SOLVE)."},"desmos":{"expression":"y = 4x^3 - 3x + 0.4","instructions":"Type the equation into the input bar to find the x-intercepts."},"description":"Solve the cubic equation $4x^3 - 3x + 0.4 = 0$ using a polynomial solver tool."}]},{"type":"text","order":2,"title":"Find angles for $x_1$","content":"Let's find the values of $\\theta$ for our first root, $x_1 \\approx 0.1367$:\n$$\\sin\\theta \\approx 0.1367$$\n\nFinding the principal value in the range $0 \\leq \\theta \\leq 180^{\\circ}$:\n$$\\theta_1 = \\sin^{-1}(0.1367) \\approx 7.9^{\\circ}$$\n\nUsing the symmetric property of the sine function ($\\sin(180^{\\circ} - \\theta) = \\sin\\theta$), we find the second angle:\n$$\\theta_2 = 180^{\\circ} - 7.855^{\\circ} \\approx 172.1^{\\circ}$$"},{"type":"text","order":3,"title":"Find angles for $x_2$","content":"Now we find the values of $\\theta$ for the second root, $x_2 \\approx 0.7895$:\n$$\\sin\\theta \\approx 0.7895$$\n\nFinding the principal value in our range:\n$$\\theta_3 = \\sin^{-1}(0.7895) \\approx 52.1^{\\circ}$$\n\nUsing the symmetry of sine again to find the second angle in this range:\n$$\\theta_4 = 180^{\\circ} - 52.14^{\\circ} \\approx 127.9^{\\circ}$$"},{"type":"text","order":4,"title":"Analyze the third root","content":"Finally, we check the third root, $x_3 \\approx -0.9262$:\n$$\\sin\\theta \\approx -0.9262$$\n\nSince $\\sin\\theta$ is negative in this case, any solution for $\\theta$ would lie in the third or fourth quadrant ($180^{\\circ} < \\theta < 360^{\\circ}$ or $-180^{\\circ} < \\theta < 0^{\\circ}$).\n\nSince our interval is strictly $0 \\leq \\theta \\leq 180^{\\circ}$ where $\\sin\\theta \\geq 0$, this root yields **no valid solutions** within the required domain.","highlights":[{"text":"0 \\leq \\theta \\leq 180","location":"specification"}]},{"type":"text","order":5,"title":"State the final solutions","content":"Combining all the valid values of $\\theta$ and rounding them to 1 decimal place, we get:\n\n$$\\theta = 7.9^{\\circ}, \\; 52.1^{\\circ}, \\; 127.9^{\\circ}, \\; 172.1^{\\circ}$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]},{"id":"graphical_intersection","label":"Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Set up the functions","content":"To solve the equation $3\\sin\\theta - 4\\sin^3\\theta = 0.4$ within the interval $0^\\circ \\leq \\theta \\leq 180^\\circ$ using a graphing calculator, we can treat both sides of the equation as separate functions:\n\n* $y_1 = 3\\sin x - 4\\sin^3 x$\n* $y_2 = 0.4$\n\nHere, we use $x$ instead of $\\theta$ to match the standard input variable of most graphing tools. We want to find the $x$-values (or $\\theta$-values) where these two curves intersect."},{"type":"text","order":1,"title":"Configure calculator settings","content":"Before plotting, ensure your graphing calculator is set to **Degree** mode because the given interval $[0, 180]$ is in degrees.\n\nSet your viewing window dimensions to capture the relevant area clearly:\n* **$x$-axis (Domain):** $X_{\\text{min}} = 0$ to $X_{\\text{max}} = 180$\n* **$y$-axis (Range):** $Y_{\\text{min}} = -1.5$ to $Y_{\\text{max}} = 1.5$ (since sine values range between $-1$ and $1$)","calculator":[{"gdc":{"expression":"y = 3*sin(x) - 4*(sin(x))^3","instructions":"1. Set the angle measurement of your calculator to **Degrees**.\n2. Enter the Graph application and input:\n \$f1(x) = 3\\sin(x) - 4(\\sin(x))^3\$\n \$f2(x) = 0.4\$\n3. Adjust the window to \$x \\in [0, 180]\$ and \$y \\in [-1.5, 1.5]\$.\n4. Plot the functions."},"ti84":{"expression":"Y_1 = 3*sin(X) - 4*(sin(X))^3","instructions":"1. Press [MODE] and ensure **DEGREE** is highlighted.\n2. Press [Y=] and enter:\n \$Y_1 = 3\\sin(X) - 4(\\sin(X))^3\$\n \$Y_2 = 0.4\$\n3. Press [WINDOW] and set:\n \$Xmin = 0\$, \$Xmax = 180\$, \$Xscl = 30\$\n \$Ymin = -1.5\$, \$Ymax = 1.5\$, \$Yscl = 0.5\$\n4. Press [GRAPH] to display the curves."},"desmos":{"expression":"y = 3\\sin(x) - 4(\\sin(x))^3","instructions":"1. Click on the wrench icon (Settings) in the top-right corner and select **Degrees**.\n2. In the first box, type: `y = 3sin(x) - 4(sin(x))^3`\n3. In the second box, type: `y = 0.4`\n4. Set the $x$-axis range to `[0, 180]` and the $y$-axis range to `[-1.5, 1.5]` using the wrench menu."},"description":"Plot the curves on a graphing utility to find their intersection points in degree mode."}]},{"type":"text","order":2,"title":"Find the intersections","content":"On your graphing display, you will see a wave-like curve ($y_1$) intersecting a horizontal line ($y_2$) at four distinct points within the domain $[0, 180]$.\n\nUse your calculator's **Intersection** tool to locate the $x$-coordinates of these points.","calculator":[{"gdc":{"instructions":"1. Go to Menu / Analyze Graph / **Intersection**.\n2. Click to the left and right of each intersection point to find its coordinates.\n3. Note down each of the four $x$-values."},"ti84":{"instructions":"1. Press [2nd] [TRACE] to open the **CALCULATE** menu.\n2. Select **5: intersect**.\n3. Move the cursor near the first intersection point, then press [ENTER] for the First curve, Second curve, and Guess.\n4. Repeat this process for all four intersection points across the interval \$[0, 180]\$."},"desmos":{"instructions":"1. Click directly on the intersection points displayed on the graph.\n2. Desmos will highlight the points and display their coordinates: \$(x, y)\$."},"description":"Use the intersection finder tool to retrieve the coordinate values."}]},{"type":"text","order":3,"title":"State the final solutions","content":"From the calculator, the $x$-coordinates (representing $\\theta$) of the four intersection points are:\n\n* $\\theta_1 \\approx 7.859^\\circ$\n* $\\theta_2 \\approx 52.141^\\circ$\n* $\\theta_3 \\approx 127.859^\\circ$\n* $\\theta_4 \\approx 172.141^\\circ$\n\nRounding these to $1$ decimal place as requested:\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$"}]}],"generatedAt":"2026-06-09T10:58:22.214Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"1ba3f6ca-faaa-4fd7-ac2a-9a95f1605f51:direct_sine_substitution","label":"(a) Direct Sine Substitution","steps":[{"type":"text","order":0,"title":"Use addition formula","content":"To show the identity for $\\sin 3\\theta$, we begin by using the addition formula for sine:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nBy writing $3\\theta$ as $2\\theta + \\theta$, we let $A = 2\\theta$ and $B = \\theta$ to expand the expression:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos \\theta + \\cos 2\\theta \\sin \\theta$$\n\nThis earns the first mark M1 for a correct compound angle expansion."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sin 2\\theta = \\htmlData{anchor=sin2-sub}{\\color{@sky}{2\\sin\\theta\\cos\\theta}}$"},{"latex":"$$\\cos 2\\theta = \\htmlData{anchor=cos2-sub}{\\color{@amber}{1 - 2\\sin^2\\theta}}$"},{"latex":"$$\\sin(2\\theta + \\theta) = \\left(\\htmlData{anchor=sin2-term}{\\color{@sky}{2\\sin\\theta\\cos\\theta}}\\right)\\cos\\theta + \\left(\\htmlData{anchor=cos2-term}{\\color{@amber}{1 - 2\\sin^2\\theta}}\\right)\\sin\\theta$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sin2-sub"},{"color":"amber","style":"text-color","anchor":"cos2-sub"}],"connections":[{"to":"sin2-term","from":"sin2-sub","color":"sky","label":"substitute","style":"solid"},{"to":"cos2-term","from":"cos2-sub","color":"amber","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute double-angle formulae","description":"Next, we substitute the double-angle identities. Following the Direct Sine Substitution method, we replace $\\sin 2\\theta$ with $2\\sin\\theta\\cos\\theta$ and $\\cos 2\\theta$ directly with $1 - 2\\sin^2\\theta$."},{"type":"text","order":2,"title":"Convert remaining cosines","content":"Let's simplify the expression we obtained:\n\n$$\\sin 3\\theta = 2\\sin\\theta\\cos^2\\theta + \\sin\\theta - 2\\sin^3\\theta$$\n\nTo write the entire equation in terms of $\\sin\\theta$, we convert the remaining $\\cos^2\\theta$ term using the fundamental identity $\\cos^2\\theta = 1 - \\sin^2\\theta$:\n\n$$\\sin 3\\theta = 2\\sin\\theta(1 - \\sin^2\\theta) + \\sin\\theta - 2\\sin^3\\theta$$"},{"type":"text","order":3,"title":"Expand and simplify","content":"Now, we expand the brackets and collect the like terms to arrive at our final result:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta - 2\\sin^3\\theta + \\sin\\theta - 2\\sin^3\\theta \\\\ &= 3\\sin\\theta - 4\\sin^3\\theta \\end{aligned}$$\n\nThis completes the proof of the identity and earns the final mark A1."},{"type":"text","order":4,"title":"Recognise the equation form","content":"For Part 2, we are given the equation:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\nRecognising that the left-hand side is exactly the identity we just proved for $\\sin 3\\theta$, we can rewrite the equation as:\n\n$$\\sin 3\\theta = 0.4$$\n\nThis step earns the first mark M1."},{"type":"text","order":5,"title":"Find intervals and solve","content":"Since $0^\\circ \\leq \\theta \\leq 180^\\circ$, the interval for $3\\theta$ is:\n\n$$0^\\circ \\leq 3\\theta \\leq 540^\\circ$$\n\nWe find the principal value and secondary values in this interval:\n\n* **First quadrant:** $3\\theta \\approx 23.578^\\circ$\n* **Second quadrant:** $3\\theta = 180^\\circ - 23.578^\\circ \\approx 156.422^\\circ$\n* **Next cycle (add $360^\\circ$):**\n * $3\\theta = 23.578^\\circ + 360^\\circ \\approx 383.578^\\circ$\n * $3\\theta = 156.422^\\circ + 360^\\circ \\approx 516.422^\\circ$\n\nFinding these values for $3\\theta$ earns another mark M1."},{"type":"text","order":6,"title":"Divide to find theta","content":"Now, we divide each value of $3\\theta$ by $3$ to solve for $\\theta$:\n\n* $\\theta = \\frac{23.578^\\circ}{3} \\approx 7.9^\\circ$\n* $\\theta = \\frac{156.422^\\circ}{3} \\approx 52.1^\\circ$\n* $\\theta = \\frac{383.578^\\circ}{3} \\approx 127.9^\\circ$\n* $\\theta = \\frac{516.422^\\circ}{3} \\approx 172.1^\\circ$\n\nDividing by 3 earns M1, and giving the final answers to 1 decimal place earns A1:\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$"}]},{"id":"1ba3f6ca-faaa-4fd7-ac2a-9a95f1605f51:general_cosine_substitution","label":"(a) General Cosine Substitution","steps":[{"type":"text","order":0,"title":"Apply compound angle formula","content":"We begin by writing $\\sin 3\\theta$ as $\\sin(2\\theta + \\theta)$ and using the compound angle formula for sine:\n\n$$\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$$\n\nApplying this with $A = 2\\theta$ and $B = \\theta$ gives:\n\n$$\\sin(2\\theta + \\theta) = \\sin 2\\theta \\cos\\theta + \\cos 2\\theta \\sin\\theta$$\n\nThis key step earns the first method mark M1 on the markscheme."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Identities: } \\htmlData{anchor=sin2-id}{\\color{@sky}{\\sin 2\\theta = 2\\sin\\theta\\cos\\theta}}, \\quad \\htmlData{anchor=cos2-id}{\\color{@amber}{\\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta}}$"},{"latex":"$$\\sin 3\\theta = \\htmlData{anchor=sin2-sub}{\\color{@sky}{\\sin 2\\theta}} \\cos\\theta + \\htmlData{anchor=cos2-sub}{\\color{@amber}{\\cos 2\\theta}} \\sin\\theta$"},{"latex":"$$\\phantom{\\sin 3\\theta} = (\\color{@sky}{2\\sin\\theta\\cos\\theta})\\cos\\theta + (\\color{@amber}{\\cos^2\\theta - \\sin^2\\theta})\\sin\\theta$"}],"highlights":[{"color":"sky","style":"box","anchor":"sin2-sub"},{"color":"amber","style":"box","anchor":"cos2-sub"}],"connections":[{"to":"sin2-sub","from":"sin2-id","color":"sky","label":"substitute","style":"solid"},{"to":"cos2-sub","from":"cos2-id","color":"amber","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute double-angle formulae","description":"We now substitute the double-angle formulae. For $\\sin 2\\theta$, we use the standard identity. For $\\cos 2\\theta$, we use the general cosine double-angle identity $\\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta$."},{"type":"text","order":2,"title":"Group the terms","content":"Now we expand the brackets and group the terms containing $\\cos^2\\theta$:\n\n$$\\begin{aligned} \\sin 3\\theta &= 2\\sin\\theta\\cos^2\\theta + \\sin\\theta\\cos^2\\theta - \\sin^3\\theta \\\\ &= 3\\sin\\theta\\cos^2\\theta - \\sin^3\\theta \\end{aligned}$$"},{"type":"text","order":3,"title":"Convert cosine to sine","content":"To express everything solely in terms of $\\sin\\theta$, we use the Pythagorean identity $\\cos^2\\theta = 1 - \\sin^2\\theta$ to substitute for the remaining $\\cos^2\\theta$ term:\n\n$$\\begin{aligned} \\sin 3\\theta &= 3\\sin\\theta(1 - \\sin^2\\theta) - \\sin^3\\theta \\\\ &= 3\\sin\\theta - 3\\sin^3\\theta - \\sin^3\\theta \\\\ &= 3\\sin\\theta - 4\\sin^3\\theta \\end{aligned}$$\n\nThis completes the proof and earns the final accuracy mark A1."},{"type":"text","order":4,"title":"Recognise the equation form","content":"For Part 2, we must solve the equation:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\nSubstituting the identity $\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$ simplifies this equation to:\n\n$$\\sin 3\\theta = 0.4$$\n\nThis yields the first method mark M1."},{"type":"text","order":5,"title":"Find intervals and solve","content":"Given the boundary condition $0^\\circ \\leq \\theta \\leq 180^\\circ$, the interval for $3\\theta$ expands to:\n\n$$0^\\circ \\leq 3\\theta \\leq 540^\\circ$$\n\nWe determine the valid values of $3\\theta$ within this boundary:\n\n* **First quadrant:** $3\\theta \\approx 23.578^\\circ$\n* **Second quadrant:** $3\\theta = 180^\\circ - 23.578^\\circ \\approx 156.422^\\circ$\n* **Second period (add $360^\\circ$):**\n * $3\\theta \\approx 23.578^\\circ + 360^\\circ = 383.578^\\circ$\n * $3\\theta \\approx 156.422^\\circ + 360^\\circ = 516.422^\\circ$\n\nCorrectly identifying these angles earns the next method mark M1."},{"type":"text","order":6,"title":"Divide to find theta","content":"Finally, we divide each angle by $3$ to find the values for $\\theta$:\n\n* $\\theta = \\frac{23.578^\\circ}{3} \\approx 7.9^\\circ$\n* $\\theta = \\frac{156.422^\\circ}{3} \\approx 52.1^\\circ$\n* $\\theta = \\frac{383.578^\\circ}{3} \\approx 127.9^\\circ$\n* $\\theta = \\frac{516.422^\\circ}{3} \\approx 172.1^\\circ$\n\nDividing by 3 earns M1, and writing the solutions to 1 decimal place earns the final accuracy mark A1:\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$"}]},{"id":"6b66702a-1a9d-459e-b0b5-e98a379d4cd5:triple_angle_identity","label":"(b) Triple Angle Identity","steps":[{"type":"text","order":0,"title":"Apply triple angle identity","content":"Recall the triple-angle identity for sine:\n\n$$\\sin 3\\theta = 3\\sin\\theta - 4\\sin^3\\theta$$\n\nWe can see that the left-hand side of our equation is exactly the expansion of this identity. Therefore, we can rewrite the equation:\n\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\nas:\n\n$$\\sin 3\\theta = 0.4$$\n\nThis earns the first method mark (**M1**).","highlights":[{"text":"3\\sin\\theta - 4\\sin^3\\theta = 0.4","location":"specification"}]},{"type":"text","order":1,"title":"Determine the interval","content":"Our original interval for $\\theta$ is:\n\n$$0^\\circ \\le \\theta \\le 180^\\circ$$\n\nSince we are solving for $3\\theta$, we must adjust the interval by multiplying all parts by $3$:\n\n$$0^\\circ \\le 3\\theta \\le 540^\\circ$$","highlights":[{"text":"0 \\leq \\theta \\leq 180","location":"specification"}]},{"type":"text","order":2,"title":"Find values for 3θ","content":"Next, we find all possible values of $3\\theta$ within our adjusted interval $[0^\\circ, 540^\\circ]$.\n\n1. Find the principal value using the inverse sine function:\n $$3\\theta_1 = \\arcsin(0.4) \\approx 23.578^\\circ$$\n\n2. Use the symmetry of the sine graph ($\\sin(180^\\circ - x) = \\sin x$) to find the second solution:\n $$3\\theta_2 = 180^\\circ - 23.578^\\circ = 156.422^\\circ$$\n\n3. Add $360^\\circ$ (the period of sine) to find subsequent solutions within $[0^\\circ, 540^\\circ]$:\n $$3\\theta_3 = 23.578^\\circ + 360^\\circ = 383.578^\\circ$$\n $$3\\theta_4 = 156.422^\\circ + 360^\\circ = 516.422^\\circ$$\n\nThese four values for $3\\theta$ earn the second method mark (**M1**)."},{"type":"text","order":3,"title":"Solve for θ","content":"Now, divide each of these values by $3$ to find the corresponding values for $\\theta$ (**M1**):\n\n$$\\begin{aligned} \\theta_1 &= \\frac{23.578^\\circ}{3} \\approx 7.859^\\circ \\\\ \\theta_2 &= \\frac{156.422^\\circ}{3} \\approx 52.141^\\circ \\\\ \\theta_3 &= \\frac{383.578^\\circ}{3} \\approx 127.859^\\circ \\\\ \\theta_4 &= \\frac{516.422^\\circ}{3} \\approx 172.141^\\circ \\end{aligned}$$\n\nRounding each final answer to $1$ decimal place gives (**A1**):\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]},{"id":"6b66702a-1a9d-459e-b0b5-e98a379d4cd5:cubic_substitution","label":"(b) Substitution and Cubic Equation Solver","steps":[{"type":"text","order":0,"title":"Substitute $x = \\sin\\theta$","content":"To solve this equation using a polynomial approach, we can simplify it by introducing a substitution.\n\nLet $x = \\sin\\theta$. Substituting this into the equation:\n$$3\\sin\\theta - 4\\sin^3\\theta = 0.4$$\n\ngives us:\n$$3x - 4x^3 = 0.4$$\n\nRearranging this into the standard cubic form $ax^3 + bx^2 + cx + d = 0$ by moving all terms to one side, we get:\n$$4x^3 - 3x + 0.4 = 0$$","highlights":[{"text":"3\\sin\\theta - 4\\sin^3\\theta = 0.4","location":"specification"}]},{"type":"text","order":1,"title":"Solve the cubic equation","content":"Using a graphing calculator's polynomial solver, we can find the real roots of this cubic equation:\n$$4x^3 - 3x + 0.4 = 0$$\n\nThe roots (to 4 decimal places) are:\n$$\\begin{aligned}\nx_1 &\\approx 0.1367 \\\\\nx_2 &\\approx 0.7895 \\\\\nx_3 &\\approx -0.9262\n\\end{aligned}$$\n\nSince $x = \\sin\\theta$, we will solve $\\sin\\theta = x$ for each of these three values within the interval $0 \\leq \\theta \\leq 180^{\\circ}$.","calculator":[{"gdc":{"expression":"4x^3 - 3x + 0.4 = 0","instructions":"Go to the Equation/Solver menu. Select 'Polynomial' (F2), and set the degree to 3. Input the coefficients: a = 4, b = 0, c = -3, d = 0.4. Solve to obtain the roots."},"ti84":{"expression":"4x^3 - 3x + 0.4 = 0","instructions":"Press [APPS], select 'PlySmlt2' (Polynomial Root Finder and Simultaneous Equation Solver). Choose 'POLY ROOT FINDER'. Set Order to 3. Enter the coefficients: a3 = 4, a2 = 0, a1 = -3, a0 = 0.4. Press [GRAPH] (SOLVE)."},"desmos":{"expression":"y = 4x^3 - 3x + 0.4","instructions":"Type the equation into the input bar to find the x-intercepts."},"description":"Solve the cubic equation $4x^3 - 3x + 0.4 = 0$ using a polynomial solver tool."}]},{"type":"text","order":2,"title":"Find angles for $x_1$","content":"Let's find the values of $\\theta$ for our first root, $x_1 \\approx 0.1367$:\n$$\\sin\\theta \\approx 0.1367$$\n\nFinding the principal value in the range $0 \\leq \\theta \\leq 180^{\\circ}$:\n$$\\theta_1 = \\sin^{-1}(0.1367) \\approx 7.9^{\\circ}$$\n\nUsing the symmetric property of the sine function ($\\sin(180^{\\circ} - \\theta) = \\sin\\theta$), we find the second angle:\n$$\\theta_2 = 180^{\\circ} - 7.855^{\\circ} \\approx 172.1^{\\circ}$$"},{"type":"text","order":3,"title":"Find angles for $x_2$","content":"Now we find the values of $\\theta$ for the second root, $x_2 \\approx 0.7895$:\n$$\\sin\\theta \\approx 0.7895$$\n\nFinding the principal value in our range:\n$$\\theta_3 = \\sin^{-1}(0.7895) \\approx 52.1^{\\circ}$$\n\nUsing the symmetry of sine again to find the second angle in this range:\n$$\\theta_4 = 180^{\\circ} - 52.14^{\\circ} \\approx 127.9^{\\circ}$$"},{"type":"text","order":4,"title":"Analyze the third root","content":"Finally, we check the third root, $x_3 \\approx -0.9262$:\n$$\\sin\\theta \\approx -0.9262$$\n\nSince $\\sin\\theta$ is negative in this case, any solution for $\\theta$ would lie in the third or fourth quadrant ($180^{\\circ} < \\theta < 360^{\\circ}$ or $-180^{\\circ} < \\theta < 0^{\\circ}$).\n\nSince our interval is strictly $0 \\leq \\theta \\leq 180^{\\circ}$ where $\\sin\\theta \\geq 0$, this root yields **no valid solutions** within the required domain.","highlights":[{"text":"0 \\leq \\theta \\leq 180","location":"specification"}]},{"type":"text","order":5,"title":"State the final solutions","content":"Combining all the valid values of $\\theta$ and rounding them to 1 decimal place, we get:\n\n$$\\theta = 7.9^{\\circ}, \\; 52.1^{\\circ}, \\; 127.9^{\\circ}, \\; 172.1^{\\circ}$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]},{"id":"6b66702a-1a9d-459e-b0b5-e98a379d4cd5:graphical_intersection","label":"(b) Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Set up the functions","content":"To solve the equation $3\\sin\\theta - 4\\sin^3\\theta = 0.4$ within the interval $0^\\circ \\leq \\theta \\leq 180^\\circ$ using a graphing calculator, we can treat both sides of the equation as separate functions:\n\n* $y_1 = 3\\sin x - 4\\sin^3 x$\n* $y_2 = 0.4$\n\nHere, we use $x$ instead of $\\theta$ to match the standard input variable of most graphing tools. We want to find the $x$-values (or $\\theta$-values) where these two curves intersect."},{"type":"text","order":1,"title":"Configure calculator settings","content":"Before plotting, ensure your graphing calculator is set to **Degree** mode because the given interval $[0, 180]$ is in degrees.\n\nSet your viewing window dimensions to capture the relevant area clearly:\n* **$x$-axis (Domain):** $X_{\\text{min}} = 0$ to $X_{\\text{max}} = 180$\n* **$y$-axis (Range):** $Y_{\\text{min}} = -1.5$ to $Y_{\\text{max}} = 1.5$ (since sine values range between $-1$ and $1$)","calculator":[{"gdc":{"expression":"y = 3*sin(x) - 4*(sin(x))^3","instructions":"1. Set the angle measurement of your calculator to **Degrees**.\n2. Enter the Graph application and input:\n \$f1(x) = 3\\sin(x) - 4(\\sin(x))^3\$\n \$f2(x) = 0.4\$\n3. Adjust the window to \$x \\in [0, 180]\$ and \$y \\in [-1.5, 1.5]\$.\n4. Plot the functions."},"ti84":{"expression":"Y_1 = 3*sin(X) - 4*(sin(X))^3","instructions":"1. Press [MODE] and ensure **DEGREE** is highlighted.\n2. Press [Y=] and enter:\n \$Y_1 = 3\\sin(X) - 4(\\sin(X))^3\$\n \$Y_2 = 0.4\$\n3. Press [WINDOW] and set:\n \$Xmin = 0\$, \$Xmax = 180\$, \$Xscl = 30\$\n \$Ymin = -1.5\$, \$Ymax = 1.5\$, \$Yscl = 0.5\$\n4. Press [GRAPH] to display the curves."},"desmos":{"expression":"y = 3\\sin(x) - 4(\\sin(x))^3","instructions":"1. Click on the wrench icon (Settings) in the top-right corner and select **Degrees**.\n2. In the first box, type: `y = 3sin(x) - 4(sin(x))^3`\n3. In the second box, type: `y = 0.4`\n4. Set the $x$-axis range to `[0, 180]` and the $y$-axis range to `[-1.5, 1.5]` using the wrench menu."},"description":"Plot the curves on a graphing utility to find their intersection points in degree mode."}]},{"type":"text","order":2,"title":"Find the intersections","content":"On your graphing display, you will see a wave-like curve ($y_1$) intersecting a horizontal line ($y_2$) at four distinct points within the domain $[0, 180]$.\n\nUse your calculator's **Intersection** tool to locate the $x$-coordinates of these points.","calculator":[{"gdc":{"instructions":"1. Go to Menu / Analyze Graph / **Intersection**.\n2. Click to the left and right of each intersection point to find its coordinates.\n3. Note down each of the four $x$-values."},"ti84":{"instructions":"1. Press [2nd] [TRACE] to open the **CALCULATE** menu.\n2. Select **5: intersect**.\n3. Move the cursor near the first intersection point, then press [ENTER] for the First curve, Second curve, and Guess.\n4. Repeat this process for all four intersection points across the interval \$[0, 180]\$."},"desmos":{"instructions":"1. Click directly on the intersection points displayed on the graph.\n2. Desmos will highlight the points and display their coordinates: \$(x, y)\$."},"description":"Use the intersection finder tool to retrieve the coordinate values."}]},{"type":"text","order":3,"title":"State the final solutions","content":"From the calculator, the $x$-coordinates (representing $\\theta$) of the four intersection points are:\n\n* $\\theta_1 \\approx 7.859^\\circ$\n* $\\theta_2 \\approx 52.141^\\circ$\n* $\\theta_3 \\approx 127.859^\\circ$\n* $\\theta_4 \\approx 172.141^\\circ$\n\nRounding these to $1$ decimal place as requested:\n\n$$\\theta = 7.9^\\circ, \\; 52.1^\\circ, \\; 127.9^\\circ, \\; 172.1^\\circ$$"}]}],"generatedAt":"2026-06-09T10:58:22.300Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths","summaries":[{"id":"94057f61-abbc-4527-807f-573c6d2bb3ef","difficultyBand":"easy","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"1c3539bd-2b7b-42e2-96cd-ad5ecc1a9b85","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"47a37e7b-a87b-4583-8b17-28f02146334b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"51aea9f2-3e1a-41e2-b43a-1b5737a1c0fc","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"938f7bbf-f10d-4cd5-a0b0-f8e7ed8cfd61","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"04df7d63-f5cb-4485-9727-ce5baaaf8bdc","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"009035ab-b95e-4a9f-9087-d0a651547233","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0721a4ee-e4ab-467f-97e9-5fda7c60e630","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8714d6b0-9b11-45c2-b4a2-7e1eab0cf698","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"c6241b26-4040-4dd8-8d9e-7a596676f9b1","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"0de526f2-fbb9-48fa-9f0b-cff5155321f4","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2a5f70b1-aeff-4a85-a7ab-fd13723ddd4e","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"c9d3770d-2da5-4cbf-aa58-8e182df190b6","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"cef9c774-cc2f-4693-a79a-6434cadcb3f2","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"d9709422-e2ce-44dc-8fb1-5939c3d61419","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"83595e3c-e503-4372-8f45-0c7c60173098","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"38c05923-2889-40f5-996b-faab30fc7f8b","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"84369d6f-3818-4ca6-ae1e-e24ee6a51011","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"964e79ae-abef-4b55-bacc-acd8c3f7a25c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"9952e45d-1751-4c06-a231-a0b217e43cf0","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"ce19f4ad-3669-44cd-8b1b-017a0566f741","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"e888b7ed-23b7-468e-81ab-46cc13f17d40","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"f2054a5a-e003-46e0-8d27-63c056892162","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"fcd7b986-e532-410e-b16c-a93caf48f013","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true}],"title":"7.2 Double Angle Formulae","topicId":"5b9ae26d-6107-4db4-89ce-d9b6c7a49036","topicSidebar":"$L3b","topicSlug":"a-level-pure-mathematics-7-2-double-angle-formulae","topicTitle":"7.2 Double Angle Formulae"}]}],"$L3c","$L3d"]}]

7.2 Double Angle Formulae

Question 4

7.2 Double Angle Formulae Questions

7.2 Double Angle Formulae

Question 4

7.2 Double Angle Formulae Questions