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$\\sin 2\\phi$ and simplifies correctly A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correct expansion of compound angle.","code":"M"},{"id":"p2","marks":1,"description":"Substitutes double angle formulas and correctly manipulates to the target identity.","code":"A"}],"finalAnswers":["4\\cos^3 \\phi - 3\\cos \\phi"],"generatedAt":"2026-05-06T14:04:14.157Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"standard_expansion","label":"Direct Algebraic Expansion","steps":[{"type":"text","order":0,"title":"Apply compound angle identity","content":"To begin, we write $\\cos 3\\phi$ as $\\cos(2\\phi + \\phi)$.\n\nWe then use the compound angle formula for cosine:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nSubstituting $A = 2\\phi$ and $B = \\phi$ gives:\n\n$$\\cos(2\\phi + \\phi) = \\cos 2\\phi \\cos \\phi - \\sin 2\\phi \\sin \\phi$$\n\nThis earns the first method mark (M1) for correctly expanding the compound angle."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\cos(2\\phi + \\phi) = \\htmlData{anchor=cos2}{\\color{@sky}{\\cos 2\\phi}} \\cos \\phi - \\htmlData{anchor=sin2}{\\color{@amber}{\\sin 2\\phi}} \\sin \\phi$"},{"latex":"$$\\cos 2\\phi = \\htmlData{anchor=cos2-sub}{\\color{@sky}{2\\cos^2 \\phi - 1}}$"},{"latex":"$$\\sin 2\\phi = \\htmlData{anchor=sin2-sub}{\\color{@amber}{2\\sin \\phi \\cos \\phi}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"cos2"},{"color":"amber","style":"box","anchor":"sin2"}],"connections":[{"to":"cos2-sub","from":"cos2","color":"sky","label":"substitute","style":"dashed"},{"to":"sin2-sub","from":"sin2","color":"amber","label":"substitute","style":"dashed"}]},"order":1,"title":"Use double angle formulae","description":"Next, we express the double-angle terms in terms of single angles using the following standard double-angle identities from the formula booklet:\n- $\\cos 2\\phi = 2\\cos^2 \\phi - 1$\n- $\\sin 2\\phi = 2\\sin \\phi \\cos \\phi$"},{"type":"text","order":2,"title":"Substitute and expand","content":"Substituting these identities into our equation, we get:\n\n$$\\cos 3\\phi = (2\\cos^2 \\phi - 1)\\cos \\phi - (2\\sin \\phi \\cos \\phi)\\sin \\phi$$\n\nNow expand the terms:\n\n$$\\cos 3\\phi = 2\\cos^3 \\phi - \\cos \\phi - 2\\sin^2 \\phi \\cos \\phi$$"},{"type":"text","order":3,"title":"Write entirely in cosine","content":"Since we want the final expression to contain only terms with $\\cos \\phi$, we can eliminate $\\sin^2 \\phi$ using the Pythagorean identity:\n\n$$\\sin^2 \\phi = 1 - \\cos^2 \\phi$$\n\nSubstituting this in:\n\n$$\\cos 3\\phi = 2\\cos^3 \\phi - \\cos \\phi - 2(1 - \\cos^2 \\phi)\\cos \\phi$$\n\n$$\\cos 3\\phi = 2\\cos^3 \\phi - \\cos \\phi - 2\\cos \\phi + 2\\cos^3 \\phi$$"},{"type":"text","order":4,"title":"Simplify to final result","content":"Finally, group the like terms together to complete the proof:\n\n$$\\cos 3\\phi = (2\\cos^3 \\phi + 2\\cos^3 \\phi) - (\\cos \\phi + 2\\cos \\phi)$$\n\n$$\\cos 3\\phi = 4\\cos^3 \\phi - 3\\cos \\phi$$\n\nThis earns the accuracy mark (A1) for simplifying correctly to the final required form."},{"type":"text","order":5,"title":"Link to part two","content":"For Part 2, we need to solve:\n\n$$12\\cos^3 \\phi - 9\\cos \\phi = 1.5$$\n\nNotice that if we factor out $3$ from the left-hand side, we get:\n\n$$3(4\\cos^3 \\phi - 3\\cos \\phi) = 1.5$$\n\nSubstituting our identity $\\cos 3\\phi = 4\\cos^3 \\phi - 3\\cos \\phi$ into this gives:\n\n$$3\\cos 3\\phi = 1.5$$\n\n$$\\cos 3\\phi = 0.5$$"},{"type":"text","order":6,"title":"Find multiple angles","content":"Since $0^\\circ \\leq \\phi \\leq 180^\\circ$, the interval for $3\\phi$ is:\n\n$$0^\\circ \\leq 3\\phi \\leq 540^\\circ$$\n\nNow, we find all angles in this range where $\\cos 3\\phi = 0.5$:\n\n- In the first quadrant cycle: $3\\phi = 60^\\circ$\n- In the fourth quadrant cycle: $3\\phi = 360^\\circ - 60^\\circ = 300^\\circ$\n- In the next rotation (first quadrant): $3\\phi = 360^\\circ + 60^\\circ = 420^\\circ$"},{"type":"text","order":7,"title":"Solve for phi","content":"To find $\\phi$, divide each angle by $3$:\n\n$$\\phi = \\frac{60^\\circ}{3}, \\frac{300^\\circ}{3}, \\frac{420^\\circ}{3}$$\n\n$$\\phi = 20^\\circ, 100^\\circ, 140^\\circ$$\n\nAll three solutions lie within the required interval $0^\\circ \\leq \\phi \\leq 180^\\circ$."}]},{"id":"alternative_double_angle","label":"Alternative Double-Angle Substitution","steps":[{"type":"text","order":0,"title":"Apply cosine addition formula","content":"To show the identity, we start by expanding $\\cos(2\\phi + \\phi)$ using the cosine addition formula:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nSetting $A = 2\\phi$ and $B = \\phi$, we get:\n\n$$\\cos(2\\phi + \\phi) = \\cos 2\\phi \\cos \\phi - \\sin 2\\phi \\sin \\phi$$\n\nThis step earns the first mark (M1) on the markscheme."},{"type":"hyper_latex","block":{"nodes":[{"text":"Next, we substitute the alternative double-angle identity for cosine and the double-angle identity for sine:"},{"latex":"$$\\htmlData{anchor=cos2}{\\color{@sky}{\\cos 2\\phi}} = \\color{@sky}{\\cos^2\\phi - \\sin^2\\phi}$"},{"latex":"$$\\htmlData{anchor=sin2}{\\color{@amber}{\\sin 2\\phi}} = \\color{@amber}{2\\sin\\phi\\cos\\phi}$"},{"text":"Substituting these into our expanded equation yields:"},{"latex":"$$\\cos 3\\phi = \\htmlData{anchor=cos2sub}{\\color{@sky}{(\\cos^2\\phi - \\sin^2\\phi)}}\\cos\\phi - \\htmlData{anchor=sin2sub}{\\color{@amber}{(2\\sin\\phi\\cos\\phi)}}\\sin\\phi$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"cos2"},{"color":"amber","style":"text-color","anchor":"sin2"}],"connections":[{"to":"cos2sub","from":"cos2","color":"sky","label":"substitute","style":"solid"},{"to":"sin2sub","from":"sin2","color":"amber","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute double-angle formulas","description":"We replace the double-angle terms with single-angle equivalent terms."},{"type":"text","order":2,"title":"Simplify the expression","content":"Now, we expand the brackets to combine the terms:\n\n$$\\cos 3\\phi = \\cos^3\\phi - \\sin^2\\phi\\cos\\phi - 2\\sin^2\\phi\\cos\\phi$$\n\nCombining the like terms $-\\sin^2\\phi\\cos\\phi$ and $-2\\sin^2\\phi\\cos\\phi$, we obtain:\n\n$$\\cos 3\\phi = \\cos^3\\phi - 3\\sin^2\\phi\\cos\\phi$$"},{"type":"text","order":3,"title":"Apply Pythagorean identity","content":"To write the entire expression in terms of $\\cos\\phi$, we use the fundamental Pythagorean identity to replace $\\sin^2\\phi$:\n\n$$\\sin^2\\phi = 1 - \\cos^2\\phi$$\n\nSubstituting this into our simplified expression:\n\n$$\\begin{aligned} \\cos 3\\phi &= \\cos^3\\phi - 3(1 - \\cos^2\\phi)\\cos\\phi \\\\ &= \\cos^3\\phi - 3(\\cos\\phi - \\cos^3\\phi) \\\\ &= \\cos^3\\phi - 3\\cos\\phi + 3\\cos^3\\phi \\end{aligned}$$\n\nCombining the $\\cos^3\\phi$ terms gives us the final result:\n\n$$\\cos 3\\phi = 4\\cos^3\\phi - 3\\cos\\phi$$\n\nThis completes the proof and earns the final mark (A1)."},{"type":"text","order":4,"title":"Formulate cosine equation","content":"For Part 2, we begin by simplifying the given equation:\n\n$$12\\cos^3 \\phi - 9\\cos \\phi = 1.5$$\n\nFactor out $3$ from the left side of the equation:\n\n$$3(4\\cos^3 \\phi - 3\\cos \\phi) = 1.5$$\n\nSince $4\\cos^3 \\phi - 3\\cos \\phi = \\cos 3\\phi$, we can substitute it in to get:\n\n$$3\\cos 3\\phi = 1.5$$\n\n$$\\cos 3\\phi = 0.5$$"},{"type":"text","order":5,"title":"Solve for angle values","content":"Since $0^\\circ \\leq \\phi \\leq 180^\\circ$, we multiply the boundaries by $3$ to find the range for $3\\phi$:\n\n$$0^\\circ \\leq 3\\phi \\leq 540^\\circ$$\n\nWithin this interval, find all solutions for $\\cos 3\\phi = 0.5$:\n\n$$3\\phi = 60^\\circ, 300^\\circ, 420^\\circ$$"},{"type":"text","order":6,"title":"Calculate final answers","content":"Divide each angle by $3$ to solve for $\\phi$:\n\n$$\\phi = \\frac{60^\\circ}{3} = 20^\\circ$$\n\n$$\\phi = \\frac{300^\\circ}{3} = 100^\\circ$$\n\n$$\\phi = \\frac{420^\\circ}{3} = 140^\\circ$$\n\nThus, our solutions are $20^\\circ$, $100^\\circ$, and $140^\\circ$."}]}],"generatedAt":"2026-06-09T09:06:10.520Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"81295664-4da6-4d8b-82b8-9cc991673af9","label":"b","content":"Solve, for $0 \\leq \\phi \\leq 180$, the equation,\n\n$$12\\cos^3 \\phi - 9\\cos \\phi = 1.5$$\n\nGive your answers to 1 decimal place where necessary.","markscheme":"- Recognises LHS as $3 \\cos 3\\phi = 1.5$, leading to $\\cos 3\\phi = 0.5$ M1\n- Finds values for $3\\phi$ in the range $[0, 540]$: $60, 300, 420$ M1\n- Divides by 3 to solve for $\\phi$ M1\n- Final answers: $20, 100, 140$ A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Recognises the expression is 3 times the identity from part (a) and finds cos 3phi = 0.5.","code":"M"},{"id":"p2","marks":1,"description":"Finds multiple solutions for 3phi.","code":"M"},{"id":"p3","marks":1,"description":"Divides through by 3.","code":"M"},{"id":"p4","marks":1,"description":"Correct final values for phi.","code":"A"}],"finalAnswers":["20, 100, 140"],"generatedAt":"2026-05-06T14:04:14.157Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"triple_angle_identity","label":"Using Triple Angle Identity","steps":[{"type":"text","order":0,"title":"Recall triple angle identity","content":"Recall the standard triple angle identity for cosine:\n\n$$\\cos(3\\phi) = 4\\cos^3 \\phi - 3\\cos \\phi$$\n\nOur equation is:\n\n$$12\\cos^3 \\phi - 9\\cos \\phi = 1.5$$\n\nNotice that the left-hand side is exactly three times the identity:\n\n$$3(4\\cos^3 \\phi - 3\\cos \\phi) = 1.5$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3\\htmlData{anchor=cos3phi}{\\color{@sky}{\\cos(3\\phi)}} = 1.5$"},{"latex":"$$\\htmlData{anchor=cos3phi-isolated}{\\color{@sky}{\\cos(3\\phi)}} = \\frac{1.5}{3} = 0.5$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"cos3phi"},{"color":"sky","style":"text-color","anchor":"cos3phi-isolated"}],"connections":[{"to":"cos3phi-isolated","from":"cos3phi","color":"sky","label":"divide by 3","style":"solid"}]},"order":1,"title":"Simplify the equation","description":"We substitute $\\cos(3\\phi)$ into our equation and then divide both sides by 3 to isolate it."},{"type":"text","order":2,"title":"Adjust the interval","content":"Our original range is given as:\n\n$$0^\\circ \\le \\phi \\le 180^\\circ$$\n\nSince our trigonometric function argument is $3\\phi$, we multiply the boundaries of our interval by $3$ to find the range for $3\\phi$:\n\n$$0^\\circ \\le 3\\phi \\le 540^\\circ$$"},{"type":"text","order":3,"title":"Find values for 3φ","content":"Now we solve $\\cos(3\\phi) = 0.5$ within the interval $[0^\\circ, 540^\\circ]$.\n\nThe principal value is:\n\n$$3\\phi = \\arccos(0.5) = 60^\\circ$$\n\nUsing the symmetry of the cosine graph, other solutions in the range $[0^\\circ, 360^\\circ]$ and its periodic extension $[360^\\circ, 540^\\circ]$ are:\n\n$$\\begin{aligned} 3\\phi &= 60^\\circ \\\\ 3\\phi &= 360^\\circ - 60^\\circ = 300^\\circ \\\\ 3\\phi &= 360^\\circ + 60^\\circ = 420^\\circ \\end{aligned}$$\n\n(Note: The next possible value, $540^\\circ - 60^\\circ = 480^\\circ$, is out of the interval for this specific ratio since $\\cos(480^\\circ) = -0.5$)"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3\\phi = \\htmlData{anchor=val1}{\\color{@pink}{60^\\circ}}, \\; \\htmlData{anchor=val2}{\\color{@purple}{300^\\circ}}, \\; \\htmlData{anchor=val3}{\\color{@teal}{420^\\circ}}$"},{"latex":"$$\\phi = \\htmlData{anchor=ans1}{\\color{@pink}{20^\\circ}}, \\; \\htmlData{anchor=ans2}{\\color{@purple}{100^\\circ}}, \\; \\htmlData{anchor=ans3}{\\color{@teal}{140^\\circ}}$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"val1"},{"color":"purple","style":"text-color","anchor":"val2"},{"color":"teal","style":"text-color","anchor":"val3"}],"connections":[{"to":"ans1","from":"val1","color":"pink","label":"divide by 3","style":"solid"},{"to":"ans2","from":"val2","color":"purple","label":"divide by 3","style":"solid"},{"to":"ans3","from":"val3","color":"teal","label":"divide by 3","style":"solid"}]},"order":4,"title":"Solve for φ","description":"Divide each value of $3\\phi$ by 3 to find the values of $\\phi$."}]},{"id":"cubic_substitution","label":"Substitution and Cubic Solver","steps":[{"type":"text","order":0,"title":"Substitute and rearrange","content":"To solve the equation $12\\cos^3 \\phi - 9\\cos \\phi = 1.5$, we begin by substituting $u = \\cos \\phi$ where $-1 \\le u \\le 1$:\n\n$$12u^3 - 9u = 1.5$$\n\nTo simplify this into a standard cubic polynomial form, we divide the entire equation by $1.5$ (or multiply by $\\frac{2}{3}$):\n\n$$8u^3 - 6u = 1$$\n\nSubtracting $1$ from both sides gives us the cubic polynomial equation:\n\n$$8u^3 - 6u - 1 = 0$$"},{"type":"text","order":1,"title":"Solve the cubic equation","content":"We can solve the cubic equation $8u^3 - 6u - 1 = 0$ using a graphing calculator or equation solver tool.\n\nLet's find the decimal values of the three real roots for $u$:\n\n$$u_1 \\approx 0.93969$$\n$$u_2 \\approx -0.17365$$\n$$u_3 \\approx -0.76604$$","calculator":[{"gdc":{"instructions":"Go to Equation mode (EQUA), select Polynomial (F2), choose Degree 3 (F2), enter coefficients: a = 8, b = 0, c = -6, d = -1, and solve."},"ti84":{"instructions":"Press [APPS], select 'PlySmlt2', then 'Poly Root Finder'. Set Order to 3. Enter the coefficients: a3 = 8, a2 = 0, a1 = -6, a0 = -1. Press [SOLVE] (graph key)."},"desmos":{"expression":"y = 8x^3 - 6x - 1","instructions":"Plot the curve y = 8x^3 - 6x - 1 and find the x-intercepts."},"description":"Solve the cubic polynomial $8u^3 - 6u - 1 = 0$ using your calculator's polynomial solver tool or by graphing the function."}]},{"type":"text","order":2,"title":"Find the angles","content":"Now, we convert back to $\\phi$ using the inverse cosine function, $\\phi = \\arccos(u)$.\n\nSince the domain is $0^\\circ \\le \\phi \\le 180^\\circ$, we only look for the principal value of each root:\n\n* For $u_1 \\approx 0.93969$:\n $$\\phi_1 = \\arccos(0.93969) \\approx 20.0^\\circ$$\n\n* For $u_2 \\approx -0.17365$:\n $$\\phi_2 = \\arccos(-0.17365) \\approx 100.0^\\circ$$\n\n* For $u_3 \\approx -0.76604$:\n $$\\phi_3 = \\arccos(-0.76604) \\approx 140.0^\\circ$$"},{"type":"text","order":3,"title":"State the final solutions","content":"The solutions to the equation within the range $0^\\circ \\leq \\phi \\leq 180^\\circ$ are:\n\n$$\\phi = 20^\\circ, \\; 100^\\circ, \\; 140^\\circ$$"}]},{"id":"graphical","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the functions","content":"To solve the equation $12\\cos^3 \\phi - 9\\cos \\phi = 1.5$ graphically, we can treat each side of the equation as a separate function and find their points of intersection.\n\nLet's define:\n$$y_1 = 12\\cos^3 \\phi - 9\\cos \\phi$$\n$$y_2 = 1.5$$\n\nWe want to find all values of $\\phi$ where $y_1 = y_2$ within the given interval $0^\\circ \\le \\phi \\le 180^\\circ$."},{"type":"text","order":1,"title":"Set up your calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode, since the interval $[0, 180]$ is specified in degrees. Next, set an appropriate viewing window to clearly see the intersections:\n- **X-axis (representing $\\phi$):** Minimum of $0$ to a maximum of $180$ with a scale of $10$ or $30$.\n- **Y-axis:** Minimum of $-4$ to a maximum of $4$ to easily capture both functions.","calculator":[{"gdc":{"expression":"f1(x) = 12(cos(x))^3 - 9cos(x), f2(x) = 1.5","instructions":"1. Ensure the angle setting is in **Degrees**.\n2. In the Graph menu, enter:\n f1(x) = 12(cos(x))^3 - 9cos(x)\n f2(x) = 1.5\n3. Adjust the window to: x in [0, 180] and y in [-4, 4].\n4. Plot the functions."},"ti84":{"expression":"Y_1 = 12(\\cos(X))^3 - 9\\cos(X), Y_2 = 1.5","instructions":"1. Press [MODE] and ensure **DEGREE** is highlighted. Press [ENTER].\n2. Press [Y=] and enter:\n Y_1 = 12(cos(X))^3 - 9cos(X)\n Y_2 = 1.5\n3. Press [WINDOW] and set:\n Xmin = 0\n Xmax = 180\n Xscl = 30\n Ymin = -4\n Ymax = 4\n4. Press [GRAPH]."},"desmos":{"expression":"y = 12(\\cos(x\\cdot\\frac{\\pi}{180}))^3 - 9\\cos(x\\cdot\\frac{\\pi}{180}), y = 1.5","instructions":"1. Click on the wrench icon (Graph Settings) in the top right.\n2. Change the angle mode from **Radians** to **Degrees**.\n3. Set the axes limits: x-axis from 0 to 180, y-axis from -4 to 4.\n4. Enter the equations into the input bar."},"description":"Configure the graphing functions and window settings on your calculator."}]},{"type":"text","order":2,"title":"Find the intersections","content":"Looking at the graphed curves, you will see three points of intersection in the interval $[0, 180]$.\n\nWe can use the calculator's intersection tool to find the exact $x$-coordinates (which represent our angle $\\phi$):\n- First intersection: $\\phi = 20^\\circ$\n- Second intersection: $\\phi = 100^\\circ$\n- Third intersection: $\\phi = 140^\\circ$","calculator":[{"gdc":{"instructions":"1. Open the **Analyze Graph** tool (or press F5 for G-Solv depending on your model) and select **Intersection**.\n2. Click or select the boundaries to find each intersection point. The results are x = 20, x = 100, and x = 140."},"ti84":{"instructions":"1. Press [2nd] [CALC] and select **5: intersect**.\n2. Move the cursor near the first intersection point on the left and press [ENTER] three times. The calculator displays X = 20.\n3. Repeat this process for the other two intersection points to get X = 100 and X = 140."},"desmos":{"instructions":"1. Click directly on the grey intersection points where the two plotted lines cross.\n2. The coordinates will pop up: (20, 1.5), (100, 1.5), and (140, 1.5)."},"description":"Use the intersection calculation tool to find the coordinate values."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The solutions to the equation within the interval $0^\\circ \\le \\phi \\le 180^\\circ$ are:\n\n$$\\phi = 20^\\circ, \\; 100^\\circ, \\; 140^\\circ$$"}]}],"generatedAt":"2026-06-09T09:08:55.405Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"934ee04c-8c35-4aea-842a-9d722bcb44ca:standard_expansion","label":"(a) Direct Algebraic Expansion","steps":[{"type":"text","order":0,"title":"Apply compound angle identity","content":"To begin, we write $\\cos 3\\phi$ as $\\cos(2\\phi + \\phi)$.\n\nWe then use the compound angle formula for cosine:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nSubstituting $A = 2\\phi$ and $B = \\phi$ gives:\n\n$$\\cos(2\\phi + \\phi) = \\cos 2\\phi \\cos \\phi - \\sin 2\\phi \\sin \\phi$$\n\nThis earns the first method mark (M1) for correctly expanding the compound angle."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\cos(2\\phi + \\phi) = \\htmlData{anchor=cos2}{\\color{@sky}{\\cos 2\\phi}} \\cos \\phi - \\htmlData{anchor=sin2}{\\color{@amber}{\\sin 2\\phi}} \\sin \\phi$"},{"latex":"$$\\cos 2\\phi = \\htmlData{anchor=cos2-sub}{\\color{@sky}{2\\cos^2 \\phi - 1}}$"},{"latex":"$$\\sin 2\\phi = \\htmlData{anchor=sin2-sub}{\\color{@amber}{2\\sin \\phi \\cos \\phi}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"cos2"},{"color":"amber","style":"box","anchor":"sin2"}],"connections":[{"to":"cos2-sub","from":"cos2","color":"sky","label":"substitute","style":"dashed"},{"to":"sin2-sub","from":"sin2","color":"amber","label":"substitute","style":"dashed"}]},"order":1,"title":"Use double angle formulae","description":"Next, we express the double-angle terms in terms of single angles using the following standard double-angle identities from the formula booklet:\n- $\\cos 2\\phi = 2\\cos^2 \\phi - 1$\n- $\\sin 2\\phi = 2\\sin \\phi \\cos \\phi$"},{"type":"text","order":2,"title":"Substitute and expand","content":"Substituting these identities into our equation, we get:\n\n$$\\cos 3\\phi = (2\\cos^2 \\phi - 1)\\cos \\phi - (2\\sin \\phi \\cos \\phi)\\sin \\phi$$\n\nNow expand the terms:\n\n$$\\cos 3\\phi = 2\\cos^3 \\phi - \\cos \\phi - 2\\sin^2 \\phi \\cos \\phi$$"},{"type":"text","order":3,"title":"Write entirely in cosine","content":"Since we want the final expression to contain only terms with $\\cos \\phi$, we can eliminate $\\sin^2 \\phi$ using the Pythagorean identity:\n\n$$\\sin^2 \\phi = 1 - \\cos^2 \\phi$$\n\nSubstituting this in:\n\n$$\\cos 3\\phi = 2\\cos^3 \\phi - \\cos \\phi - 2(1 - \\cos^2 \\phi)\\cos \\phi$$\n\n$$\\cos 3\\phi = 2\\cos^3 \\phi - \\cos \\phi - 2\\cos \\phi + 2\\cos^3 \\phi$$"},{"type":"text","order":4,"title":"Simplify to final result","content":"Finally, group the like terms together to complete the proof:\n\n$$\\cos 3\\phi = (2\\cos^3 \\phi + 2\\cos^3 \\phi) - (\\cos \\phi + 2\\cos \\phi)$$\n\n$$\\cos 3\\phi = 4\\cos^3 \\phi - 3\\cos \\phi$$\n\nThis earns the accuracy mark (A1) for simplifying correctly to the final required form."},{"type":"text","order":5,"title":"Link to part two","content":"For Part 2, we need to solve:\n\n$$12\\cos^3 \\phi - 9\\cos \\phi = 1.5$$\n\nNotice that if we factor out $3$ from the left-hand side, we get:\n\n$$3(4\\cos^3 \\phi - 3\\cos \\phi) = 1.5$$\n\nSubstituting our identity $\\cos 3\\phi = 4\\cos^3 \\phi - 3\\cos \\phi$ into this gives:\n\n$$3\\cos 3\\phi = 1.5$$\n\n$$\\cos 3\\phi = 0.5$$"},{"type":"text","order":6,"title":"Find multiple angles","content":"Since $0^\\circ \\leq \\phi \\leq 180^\\circ$, the interval for $3\\phi$ is:\n\n$$0^\\circ \\leq 3\\phi \\leq 540^\\circ$$\n\nNow, we find all angles in this range where $\\cos 3\\phi = 0.5$:\n\n- In the first quadrant cycle: $3\\phi = 60^\\circ$\n- In the fourth quadrant cycle: $3\\phi = 360^\\circ - 60^\\circ = 300^\\circ$\n- In the next rotation (first quadrant): $3\\phi = 360^\\circ + 60^\\circ = 420^\\circ$"},{"type":"text","order":7,"title":"Solve for phi","content":"To find $\\phi$, divide each angle by $3$:\n\n$$\\phi = \\frac{60^\\circ}{3}, \\frac{300^\\circ}{3}, \\frac{420^\\circ}{3}$$\n\n$$\\phi = 20^\\circ, 100^\\circ, 140^\\circ$$\n\nAll three solutions lie within the required interval $0^\\circ \\leq \\phi \\leq 180^\\circ$."}]},{"id":"934ee04c-8c35-4aea-842a-9d722bcb44ca:alternative_double_angle","label":"(a) Alternative Double-Angle Substitution","steps":[{"type":"text","order":0,"title":"Apply cosine addition formula","content":"To show the identity, we start by expanding $\\cos(2\\phi + \\phi)$ using the cosine addition formula:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nSetting $A = 2\\phi$ and $B = \\phi$, we get:\n\n$$\\cos(2\\phi + \\phi) = \\cos 2\\phi \\cos \\phi - \\sin 2\\phi \\sin \\phi$$\n\nThis step earns the first mark (M1) on the markscheme."},{"type":"hyper_latex","block":{"nodes":[{"text":"Next, we substitute the alternative double-angle identity for cosine and the double-angle identity for sine:"},{"latex":"$$\\htmlData{anchor=cos2}{\\color{@sky}{\\cos 2\\phi}} = \\color{@sky}{\\cos^2\\phi - \\sin^2\\phi}$"},{"latex":"$$\\htmlData{anchor=sin2}{\\color{@amber}{\\sin 2\\phi}} = \\color{@amber}{2\\sin\\phi\\cos\\phi}$"},{"text":"Substituting these into our expanded equation yields:"},{"latex":"$$\\cos 3\\phi = \\htmlData{anchor=cos2sub}{\\color{@sky}{(\\cos^2\\phi - \\sin^2\\phi)}}\\cos\\phi - \\htmlData{anchor=sin2sub}{\\color{@amber}{(2\\sin\\phi\\cos\\phi)}}\\sin\\phi$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"cos2"},{"color":"amber","style":"text-color","anchor":"sin2"}],"connections":[{"to":"cos2sub","from":"cos2","color":"sky","label":"substitute","style":"solid"},{"to":"sin2sub","from":"sin2","color":"amber","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute double-angle formulas","description":"We replace the double-angle terms with single-angle equivalent terms."},{"type":"text","order":2,"title":"Simplify the expression","content":"Now, we expand the brackets to combine the terms:\n\n$$\\cos 3\\phi = \\cos^3\\phi - \\sin^2\\phi\\cos\\phi - 2\\sin^2\\phi\\cos\\phi$$\n\nCombining the like terms $-\\sin^2\\phi\\cos\\phi$ and $-2\\sin^2\\phi\\cos\\phi$, we obtain:\n\n$$\\cos 3\\phi = \\cos^3\\phi - 3\\sin^2\\phi\\cos\\phi$$"},{"type":"text","order":3,"title":"Apply Pythagorean identity","content":"To write the entire expression in terms of $\\cos\\phi$, we use the fundamental Pythagorean identity to replace $\\sin^2\\phi$:\n\n$$\\sin^2\\phi = 1 - \\cos^2\\phi$$\n\nSubstituting this into our simplified expression:\n\n$$\\begin{aligned} \\cos 3\\phi &= \\cos^3\\phi - 3(1 - \\cos^2\\phi)\\cos\\phi \\\\ &= \\cos^3\\phi - 3(\\cos\\phi - \\cos^3\\phi) \\\\ &= \\cos^3\\phi - 3\\cos\\phi + 3\\cos^3\\phi \\end{aligned}$$\n\nCombining the $\\cos^3\\phi$ terms gives us the final result:\n\n$$\\cos 3\\phi = 4\\cos^3\\phi - 3\\cos\\phi$$\n\nThis completes the proof and earns the final mark (A1)."},{"type":"text","order":4,"title":"Formulate cosine equation","content":"For Part 2, we begin by simplifying the given equation:\n\n$$12\\cos^3 \\phi - 9\\cos \\phi = 1.5$$\n\nFactor out $3$ from the left side of the equation:\n\n$$3(4\\cos^3 \\phi - 3\\cos \\phi) = 1.5$$\n\nSince $4\\cos^3 \\phi - 3\\cos \\phi = \\cos 3\\phi$, we can substitute it in to get:\n\n$$3\\cos 3\\phi = 1.5$$\n\n$$\\cos 3\\phi = 0.5$$"},{"type":"text","order":5,"title":"Solve for angle values","content":"Since $0^\\circ \\leq \\phi \\leq 180^\\circ$, we multiply the boundaries by $3$ to find the range for $3\\phi$:\n\n$$0^\\circ \\leq 3\\phi \\leq 540^\\circ$$\n\nWithin this interval, find all solutions for $\\cos 3\\phi = 0.5$:\n\n$$3\\phi = 60^\\circ, 300^\\circ, 420^\\circ$$"},{"type":"text","order":6,"title":"Calculate final answers","content":"Divide each angle by $3$ to solve for $\\phi$:\n\n$$\\phi = \\frac{60^\\circ}{3} = 20^\\circ$$\n\n$$\\phi = \\frac{300^\\circ}{3} = 100^\\circ$$\n\n$$\\phi = \\frac{420^\\circ}{3} = 140^\\circ$$\n\nThus, our solutions are $20^\\circ$, $100^\\circ$, and $140^\\circ$."}]},{"id":"81295664-4da6-4d8b-82b8-9cc991673af9:triple_angle_identity","label":"(b) Using Triple Angle Identity","steps":[{"type":"text","order":0,"title":"Recall triple angle identity","content":"Recall the standard triple angle identity for cosine:\n\n$$\\cos(3\\phi) = 4\\cos^3 \\phi - 3\\cos \\phi$$\n\nOur equation is:\n\n$$12\\cos^3 \\phi - 9\\cos \\phi = 1.5$$\n\nNotice that the left-hand side is exactly three times the identity:\n\n$$3(4\\cos^3 \\phi - 3\\cos \\phi) = 1.5$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3\\htmlData{anchor=cos3phi}{\\color{@sky}{\\cos(3\\phi)}} = 1.5$"},{"latex":"$$\\htmlData{anchor=cos3phi-isolated}{\\color{@sky}{\\cos(3\\phi)}} = \\frac{1.5}{3} = 0.5$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"cos3phi"},{"color":"sky","style":"text-color","anchor":"cos3phi-isolated"}],"connections":[{"to":"cos3phi-isolated","from":"cos3phi","color":"sky","label":"divide by 3","style":"solid"}]},"order":1,"title":"Simplify the equation","description":"We substitute $\\cos(3\\phi)$ into our equation and then divide both sides by 3 to isolate it."},{"type":"text","order":2,"title":"Adjust the interval","content":"Our original range is given as:\n\n$$0^\\circ \\le \\phi \\le 180^\\circ$$\n\nSince our trigonometric function argument is $3\\phi$, we multiply the boundaries of our interval by $3$ to find the range for $3\\phi$:\n\n$$0^\\circ \\le 3\\phi \\le 540^\\circ$$"},{"type":"text","order":3,"title":"Find values for 3φ","content":"Now we solve $\\cos(3\\phi) = 0.5$ within the interval $[0^\\circ, 540^\\circ]$.\n\nThe principal value is:\n\n$$3\\phi = \\arccos(0.5) = 60^\\circ$$\n\nUsing the symmetry of the cosine graph, other solutions in the range $[0^\\circ, 360^\\circ]$ and its periodic extension $[360^\\circ, 540^\\circ]$ are:\n\n$$\\begin{aligned} 3\\phi &= 60^\\circ \\\\ 3\\phi &= 360^\\circ - 60^\\circ = 300^\\circ \\\\ 3\\phi &= 360^\\circ + 60^\\circ = 420^\\circ \\end{aligned}$$\n\n(Note: The next possible value, $540^\\circ - 60^\\circ = 480^\\circ$, is out of the interval for this specific ratio since $\\cos(480^\\circ) = -0.5$)"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3\\phi = \\htmlData{anchor=val1}{\\color{@pink}{60^\\circ}}, \\; \\htmlData{anchor=val2}{\\color{@purple}{300^\\circ}}, \\; \\htmlData{anchor=val3}{\\color{@teal}{420^\\circ}}$"},{"latex":"$$\\phi = \\htmlData{anchor=ans1}{\\color{@pink}{20^\\circ}}, \\; \\htmlData{anchor=ans2}{\\color{@purple}{100^\\circ}}, \\; \\htmlData{anchor=ans3}{\\color{@teal}{140^\\circ}}$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"val1"},{"color":"purple","style":"text-color","anchor":"val2"},{"color":"teal","style":"text-color","anchor":"val3"}],"connections":[{"to":"ans1","from":"val1","color":"pink","label":"divide by 3","style":"solid"},{"to":"ans2","from":"val2","color":"purple","label":"divide by 3","style":"solid"},{"to":"ans3","from":"val3","color":"teal","label":"divide by 3","style":"solid"}]},"order":4,"title":"Solve for φ","description":"Divide each value of $3\\phi$ by 3 to find the values of $\\phi$."}]},{"id":"81295664-4da6-4d8b-82b8-9cc991673af9:cubic_substitution","label":"(b) Substitution and Cubic Solver","steps":[{"type":"text","order":0,"title":"Substitute and rearrange","content":"To solve the equation $12\\cos^3 \\phi - 9\\cos \\phi = 1.5$, we begin by substituting $u = \\cos \\phi$ where $-1 \\le u \\le 1$:\n\n$$12u^3 - 9u = 1.5$$\n\nTo simplify this into a standard cubic polynomial form, we divide the entire equation by $1.5$ (or multiply by $\\frac{2}{3}$):\n\n$$8u^3 - 6u = 1$$\n\nSubtracting $1$ from both sides gives us the cubic polynomial equation:\n\n$$8u^3 - 6u - 1 = 0$$"},{"type":"text","order":1,"title":"Solve the cubic equation","content":"We can solve the cubic equation $8u^3 - 6u - 1 = 0$ using a graphing calculator or equation solver tool.\n\nLet's find the decimal values of the three real roots for $u$:\n\n$$u_1 \\approx 0.93969$$\n$$u_2 \\approx -0.17365$$\n$$u_3 \\approx -0.76604$$","calculator":[{"gdc":{"instructions":"Go to Equation mode (EQUA), select Polynomial (F2), choose Degree 3 (F2), enter coefficients: a = 8, b = 0, c = -6, d = -1, and solve."},"ti84":{"instructions":"Press [APPS], select 'PlySmlt2', then 'Poly Root Finder'. Set Order to 3. Enter the coefficients: a3 = 8, a2 = 0, a1 = -6, a0 = -1. Press [SOLVE] (graph key)."},"desmos":{"expression":"y = 8x^3 - 6x - 1","instructions":"Plot the curve y = 8x^3 - 6x - 1 and find the x-intercepts."},"description":"Solve the cubic polynomial $8u^3 - 6u - 1 = 0$ using your calculator's polynomial solver tool or by graphing the function."}]},{"type":"text","order":2,"title":"Find the angles","content":"Now, we convert back to $\\phi$ using the inverse cosine function, $\\phi = \\arccos(u)$.\n\nSince the domain is $0^\\circ \\le \\phi \\le 180^\\circ$, we only look for the principal value of each root:\n\n* For $u_1 \\approx 0.93969$:\n $$\\phi_1 = \\arccos(0.93969) \\approx 20.0^\\circ$$\n\n* For $u_2 \\approx -0.17365$:\n $$\\phi_2 = \\arccos(-0.17365) \\approx 100.0^\\circ$$\n\n* For $u_3 \\approx -0.76604$:\n $$\\phi_3 = \\arccos(-0.76604) \\approx 140.0^\\circ$$"},{"type":"text","order":3,"title":"State the final solutions","content":"The solutions to the equation within the range $0^\\circ \\leq \\phi \\leq 180^\\circ$ are:\n\n$$\\phi = 20^\\circ, \\; 100^\\circ, \\; 140^\\circ$$"}]},{"id":"81295664-4da6-4d8b-82b8-9cc991673af9:graphical","label":"(b) Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the functions","content":"To solve the equation $12\\cos^3 \\phi - 9\\cos \\phi = 1.5$ graphically, we can treat each side of the equation as a separate function and find their points of intersection.\n\nLet's define:\n$$y_1 = 12\\cos^3 \\phi - 9\\cos \\phi$$\n$$y_2 = 1.5$$\n\nWe want to find all values of $\\phi$ where $y_1 = y_2$ within the given interval $0^\\circ \\le \\phi \\le 180^\\circ$."},{"type":"text","order":1,"title":"Set up your calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode, since the interval $[0, 180]$ is specified in degrees. Next, set an appropriate viewing window to clearly see the intersections:\n- **X-axis (representing $\\phi$):** Minimum of $0$ to a maximum of $180$ with a scale of $10$ or $30$.\n- **Y-axis:** Minimum of $-4$ to a maximum of $4$ to easily capture both functions.","calculator":[{"gdc":{"expression":"f1(x) = 12(cos(x))^3 - 9cos(x), f2(x) = 1.5","instructions":"1. Ensure the angle setting is in **Degrees**.\n2. In the Graph menu, enter:\n f1(x) = 12(cos(x))^3 - 9cos(x)\n f2(x) = 1.5\n3. Adjust the window to: x in [0, 180] and y in [-4, 4].\n4. Plot the functions."},"ti84":{"expression":"Y_1 = 12(\\cos(X))^3 - 9\\cos(X), Y_2 = 1.5","instructions":"1. Press [MODE] and ensure **DEGREE** is highlighted. Press [ENTER].\n2. Press [Y=] and enter:\n Y_1 = 12(cos(X))^3 - 9cos(X)\n Y_2 = 1.5\n3. Press [WINDOW] and set:\n Xmin = 0\n Xmax = 180\n Xscl = 30\n Ymin = -4\n Ymax = 4\n4. Press [GRAPH]."},"desmos":{"expression":"y = 12(\\cos(x\\cdot\\frac{\\pi}{180}))^3 - 9\\cos(x\\cdot\\frac{\\pi}{180}), y = 1.5","instructions":"1. Click on the wrench icon (Graph Settings) in the top right.\n2. Change the angle mode from **Radians** to **Degrees**.\n3. Set the axes limits: x-axis from 0 to 180, y-axis from -4 to 4.\n4. Enter the equations into the input bar."},"description":"Configure the graphing functions and window settings on your calculator."}]},{"type":"text","order":2,"title":"Find the intersections","content":"Looking at the graphed curves, you will see three points of intersection in the interval $[0, 180]$.\n\nWe can use the calculator's intersection tool to find the exact $x$-coordinates (which represent our angle $\\phi$):\n- First intersection: $\\phi = 20^\\circ$\n- Second intersection: $\\phi = 100^\\circ$\n- Third intersection: $\\phi = 140^\\circ$","calculator":[{"gdc":{"instructions":"1. Open the **Analyze Graph** tool (or press F5 for G-Solv depending on your model) and select **Intersection**.\n2. Click or select the boundaries to find each intersection point. The results are x = 20, x = 100, and x = 140."},"ti84":{"instructions":"1. Press [2nd] [CALC] and select **5: intersect**.\n2. Move the cursor near the first intersection point on the left and press [ENTER] three times. The calculator displays X = 20.\n3. Repeat this process for the other two intersection points to get X = 100 and X = 140."},"desmos":{"instructions":"1. Click directly on the grey intersection points where the two plotted lines cross.\n2. The coordinates will pop up: (20, 1.5), (100, 1.5), and (140, 1.5)."},"description":"Use the intersection calculation tool to find the coordinate values."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The solutions to the equation within the interval $0^\\circ \\le \\phi \\le 180^\\circ$ are:\n\n$$\\phi = 20^\\circ, \\; 100^\\circ, \\; 140^\\circ$$"}]}],"generatedAt":"2026-06-09T09:09:07.922Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"cef9c774-cc2f-4693-a79a-6434cadcb3f2","revisionId":"e44e2d5d-f581-4b66-9018-6636e2384c8d","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"660d452f-49d6-4939-9567-b4cf039d955a","label":"a","content":"Solve, for $-180 \\leq x \\leq 180$, the equation,\n\n$$2 \\cos 2x + 7 \\cos x = 0$$","markscheme":"- Substitutes $\\cos 2x = 2 \\cos^2 x - 1$ M1\n- Forms quadratic: $4 \\cos^2 x + 7 \\cos x - 2 = 0$ M1\n- Factorises to $(4 \\cos x - 1)(\\cos x + 2) = 0$ M1\n- Solves $\\cos x = \\frac{1}{4}$ (rejects $\\cos x = -2$) M1\n- Final answers: $x \\approx 75.5^\\circ, -75.5^\\circ$ (to 1 d.p.) A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Attempt to substitute identity $\\cos 2x = 2 \\cos^2 x - 1$.","code":"M"},{"id":"p2","marks":1,"description":"Formation of a correct 3-term quadratic equation in $\\cos x$.","code":"M"},{"id":"p3","marks":1,"description":"Correct factorization or solution method for their quadratic equation.","code":"M"},{"id":"p4","marks":1,"description":"Identification of $\\cos x = 1/4$ and explicit or implied rejection of the other root.","code":"M"},{"id":"p5","marks":1,"description":"Final answers $x = 75.5^\\circ$ and $x = -75.5^\\circ$ (or similar rounding) provided correctly.","code":"A"}],"finalAnswers":["75.5^\\circ, -75.5^\\circ"],"generatedAt":"2026-05-06T14:04:22.546Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"algebraic_quadratic","label":"Algebraic Approach with Quadratic Formula","steps":[{"type":"text","order":0,"title":"Use double-angle identity","content":"To solve the equation, we first need to express it entirely in terms of $\\cos x$. We can do this by using the double-angle identity for cosine from our formula sheet:\n\n$$\\cos 2x = 2 \\cos^2 x - 1$$\n\nSubstituting this into the given equation:\n\n$$2(2 \\cos^2 x - 1) + 7 \\cos x = 0$$","highlights":[{"text":"2 \\cos 2x + 7 \\cos x = 0","location":"specification"}]},{"type":"text","order":1,"title":"Form a quadratic equation","content":"Now we expand and simplify the expression to form a standard quadratic equation in terms of $\\cos x$:\n\n$$4 \\cos^2 x - 2 + 7 \\cos x = 0$$\n\nRearranging the terms, we get:\n\n$$4 \\cos^2 x + 7 \\cos x - 2 = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4\\htmlData{anchor=quad-var}{\\cos^2 x} + 7\\cos x - 2 = 0$"},{"text":"This quadratic can be factorised in the same way as $4y^2 + 7y - 2 = 0$:"},{"latex":"$$(\\htmlData{anchor=fact-1}{4 \\cos x - 1})(\\htmlData{anchor=fact-2}{\\cos x + 2}) = 0$"}],"highlights":[{"color":"sky","style":"underline","anchor":"quad-var"},{"color":"sky","style":"text-color","anchor":"fact-1"},{"color":"amber","style":"text-color","anchor":"fact-2"}],"connections":[{"to":"fact-1","from":"quad-var","color":"sky","label":"factorise","style":"solid"}]},"order":2,"title":"Factorise the quadratic","description":"Factorise the quadratic expression into two linear factors."},{"type":"text","order":3,"title":"Solve for cos x","content":"Set each factor to zero to find the possible values for $\\cos x$:\n\n1. From $4 \\cos x - 1 = 0$, we have:\n $$\\cos x = \\frac{1}{4}$$\n\n2. From $\\cos x + 2 = 0$, we have:\n $$\\cos x = -2$$\n\nSince the range of the cosine function is $-1 \\leq \\cos x \\leq 1$, the equation $\\cos x = -2$ has no real solutions. We reject this case."},{"type":"text","order":4,"title":"Find the angles","content":"Now we find the values of $x$ for $\\cos x = 0.25$ within the interval $-180^\\circ \\leq x \\leq 180^\\circ$.\n\nFirst, find the principal value using your calculator in degree mode:\n\n$$x = \\cos^{-1}(0.25) \\approx 75.52^\\circ$$\n\nUsing the symmetry of the cosine graph (where $\\cos(-x) = \\cos(x)$), our second solution is:\n\n$$x = -75.52^\\circ$$\n\nRounding both answers to 1 decimal place, we obtain:\n\n$$x \\approx 75.5^\\circ \\text{ and } x \\approx -75.5^\\circ$$","calculator":[{"gdc":{"expression":"acos(0.25)","instructions":"1. Set the angle settings of your calculator to Degrees.\n2. In the run-matrix menu, enter acos(0.25)."},"ti84":{"expression":"cos⁻¹(0.25)","instructions":"1. Press [MODE] and ensure DEGREE is highlighted.\n2. Press [2nd] [COS] to get cos⁻¹(\n3. Type 0.25 and press [ENTER]."},"desmos":{"expression":"\\arccos(0.25)","instructions":"1. Click the wrench icon in the top right and select 'Degrees'.\n2. Type arccos(0.25) into the input box."},"description":"Find the inverse cosine of 0.25 in degree mode."}],"highlights":[{"text":"-180 \\leq x \\leq 180","location":"specification"}]}]},{"id":"graphical_calculator","label":"Graphical Approach using Calculator","steps":[{"type":"text","order":0,"title":"Understand the Goal","content":"We need to find the solutions to the trigonometric equation:\n\n$$2 \\cos 2x + 7 \\cos x = 0$$\n\nfor the domain $-180^\\circ \\leq x \\leq 180^\\circ$. \n\nWe will solve this graphically by defining a function:\n\n$$y = 2 \\cos 2x + 7 \\cos x$$\n\nand finding its $x$-intercepts (where $y = 0$) within our given interval.","highlights":[{"text":"-180 \\leq x \\leq 180","location":"specification"},{"text":"2 \\cos 2x + 7 \\cos x = 0","location":"specification"}]},{"type":"text","order":1,"title":"Set Up Your Calculator","content":"Before plotting, we must ensure the calculator is configured correctly:\n1. **Angle Mode**: Since the domain is given in degrees ($-180^\\circ$ to $180^\\circ$), change your calculator's angle setting to **Degree** mode.\n2. **Viewing Window**: Set the $x$-axis range to cover the given domain, and choose a suitable vertical range to see the curve's peaks and troughs clearly.\n - $X_{\\min} = -180$\n - $X_{\\max} = 180$\n - $Y_{\\min} = -10$\n - $Y_{\\max} = 10$","calculator":[{"gdc":{"expression":"2*cos(2*x) + 7*cos(x)","instructions":"1. Set the system angle unit to Degrees.\n2. Open the Graph application.\n3. Enter the function f1(x) = 2*cos(2*x) + 7*cos(x).\n4. Adjust the Window Settings: Xmin = -180, Xmax = 180, Ymin = -10, Ymax = 10."},"ti84":{"expression":"2*cos(2*X) + 7*cos(X)","instructions":"1. Press [MODE] and ensure 'DEGREE' is highlighted.\n2. Press [Y=] and enter Y1 = 2*cos(2*X) + 7*cos(X).\n3. Press [WINDOW] and set: Xmin = -180, Xmax = 180, Xscl = 30, Ymin = -10, Ymax = 10.\n4. Press [GRAPH]."},"desmos":{"expression":"y = 2*cos(2x) + 7*cos(x)","instructions":"1. Click the wrench icon (Graph Settings).\n2. Toggle the angle mode from 'Radians' to 'Degrees'.\n3. Set the X-Axis bounds from -180 to 180.\n4. Enter the equation: y = 2*cos(2x) + 7*cos(x)."},"description":"Enter the function and set the viewing window on your graphing tool."}]},{"type":"text","order":2,"title":"Locate the x-intercepts","content":"On the screen, you will observe a symmetric curve that crosses the $x$-axis twice within the window $[-180^\\circ, 180^\\circ]$. \n\nTo find these solutions, use the calculator's tool to locate the **roots** or **zeros** of the function.","calculator":[{"gdc":{"instructions":"1. Analyze Graph -> Zero.\n2. Click to select a lower bound and an upper bound around each intersection point with the x-axis."},"ti84":{"instructions":"1. Press [2nd] [TRACE] (CALC) and select '2: zero'.\n2. Move the cursor to the left of the positive intercept and press [ENTER] (Left Bound).\n3. Move the cursor to the right of the intercept and press [ENTER] (Right Bound).\n4. Press [ENTER] again to guess. \n5. Repeat this process for the negative intercept."},"desmos":{"instructions":"1. Click on the curve.\n2. Click directly on the grey dots representing the x-intercepts to display their coordinates."},"description":"Calculate the exact coordinates of the points where the curve crosses the x-axis."}]},{"type":"text","order":3,"title":"State the Final Solutions","content":"The calculator finds the two $x$-intercepts as:\n\n$$x \\approx 75.522^\\circ \\quad \\text{and} \\quad x \\approx -75.522^\\circ$$\n\nRounding to 1 decimal place as standard for angles in degrees:\n\n$$x \\approx 75.5^\\circ \\quad \\text{and} \\quad x \\approx -75.5^\\circ$$\n\nBoth values lie within the interval $[-180^\\circ, 180^\\circ]$, so they are the valid solutions."}]}],"generatedAt":"2026-06-11T03:45:08.251Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"660d452f-49d6-4939-9567-b4cf039d955a:algebraic_quadratic","label":"Algebraic Approach with Quadratic Formula","steps":[{"type":"text","order":0,"title":"Use double-angle identity","content":"To solve the equation, we first need to express it entirely in terms of $\\cos x$. We can do this by using the double-angle identity for cosine from our formula sheet:\n\n$$\\cos 2x = 2 \\cos^2 x - 1$$\n\nSubstituting this into the given equation:\n\n$$2(2 \\cos^2 x - 1) + 7 \\cos x = 0$$","highlights":[{"text":"2 \\cos 2x + 7 \\cos x = 0","location":"specification"}]},{"type":"text","order":1,"title":"Form a quadratic equation","content":"Now we expand and simplify the expression to form a standard quadratic equation in terms of $\\cos x$:\n\n$$4 \\cos^2 x - 2 + 7 \\cos x = 0$$\n\nRearranging the terms, we get:\n\n$$4 \\cos^2 x + 7 \\cos x - 2 = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4\\htmlData{anchor=quad-var}{\\cos^2 x} + 7\\cos x - 2 = 0$"},{"text":"This quadratic can be factorised in the same way as $4y^2 + 7y - 2 = 0$:"},{"latex":"$$(\\htmlData{anchor=fact-1}{4 \\cos x - 1})(\\htmlData{anchor=fact-2}{\\cos x + 2}) = 0$"}],"highlights":[{"color":"sky","style":"underline","anchor":"quad-var"},{"color":"sky","style":"text-color","anchor":"fact-1"},{"color":"amber","style":"text-color","anchor":"fact-2"}],"connections":[{"to":"fact-1","from":"quad-var","color":"sky","label":"factorise","style":"solid"}]},"order":2,"title":"Factorise the quadratic","description":"Factorise the quadratic expression into two linear factors."},{"type":"text","order":3,"title":"Solve for cos x","content":"Set each factor to zero to find the possible values for $\\cos x$:\n\n1. From $4 \\cos x - 1 = 0$, we have:\n $$\\cos x = \\frac{1}{4}$$\n\n2. From $\\cos x + 2 = 0$, we have:\n $$\\cos x = -2$$\n\nSince the range of the cosine function is $-1 \\leq \\cos x \\leq 1$, the equation $\\cos x = -2$ has no real solutions. We reject this case."},{"type":"text","order":4,"title":"Find the angles","content":"Now we find the values of $x$ for $\\cos x = 0.25$ within the interval $-180^\\circ \\leq x \\leq 180^\\circ$.\n\nFirst, find the principal value using your calculator in degree mode:\n\n$$x = \\cos^{-1}(0.25) \\approx 75.52^\\circ$$\n\nUsing the symmetry of the cosine graph (where $\\cos(-x) = \\cos(x)$), our second solution is:\n\n$$x = -75.52^\\circ$$\n\nRounding both answers to 1 decimal place, we obtain:\n\n$$x \\approx 75.5^\\circ \\text{ and } x \\approx -75.5^\\circ$$","calculator":[{"gdc":{"expression":"acos(0.25)","instructions":"1. Set the angle settings of your calculator to Degrees.\n2. In the run-matrix menu, enter acos(0.25)."},"ti84":{"expression":"cos⁻¹(0.25)","instructions":"1. Press [MODE] and ensure DEGREE is highlighted.\n2. Press [2nd] [COS] to get cos⁻¹(\n3. Type 0.25 and press [ENTER]."},"desmos":{"expression":"\\arccos(0.25)","instructions":"1. Click the wrench icon in the top right and select 'Degrees'.\n2. Type arccos(0.25) into the input box."},"description":"Find the inverse cosine of 0.25 in degree mode."}],"highlights":[{"text":"-180 \\leq x \\leq 180","location":"specification"}]}]},{"id":"660d452f-49d6-4939-9567-b4cf039d955a:graphical_calculator","label":"Graphical Approach using Calculator","steps":[{"type":"text","order":0,"title":"Understand the Goal","content":"We need to find the solutions to the trigonometric equation:\n\n$$2 \\cos 2x + 7 \\cos x = 0$$\n\nfor the domain $-180^\\circ \\leq x \\leq 180^\\circ$. \n\nWe will solve this graphically by defining a function:\n\n$$y = 2 \\cos 2x + 7 \\cos x$$\n\nand finding its $x$-intercepts (where $y = 0$) within our given interval.","highlights":[{"text":"-180 \\leq x \\leq 180","location":"specification"},{"text":"2 \\cos 2x + 7 \\cos x = 0","location":"specification"}]},{"type":"text","order":1,"title":"Set Up Your Calculator","content":"Before plotting, we must ensure the calculator is configured correctly:\n1. **Angle Mode**: Since the domain is given in degrees ($-180^\\circ$ to $180^\\circ$), change your calculator's angle setting to **Degree** mode.\n2. **Viewing Window**: Set the $x$-axis range to cover the given domain, and choose a suitable vertical range to see the curve's peaks and troughs clearly.\n - $X_{\\min} = -180$\n - $X_{\\max} = 180$\n - $Y_{\\min} = -10$\n - $Y_{\\max} = 10$","calculator":[{"gdc":{"expression":"2*cos(2*x) + 7*cos(x)","instructions":"1. Set the system angle unit to Degrees.\n2. Open the Graph application.\n3. Enter the function f1(x) = 2*cos(2*x) + 7*cos(x).\n4. Adjust the Window Settings: Xmin = -180, Xmax = 180, Ymin = -10, Ymax = 10."},"ti84":{"expression":"2*cos(2*X) + 7*cos(X)","instructions":"1. Press [MODE] and ensure 'DEGREE' is highlighted.\n2. Press [Y=] and enter Y1 = 2*cos(2*X) + 7*cos(X).\n3. Press [WINDOW] and set: Xmin = -180, Xmax = 180, Xscl = 30, Ymin = -10, Ymax = 10.\n4. Press [GRAPH]."},"desmos":{"expression":"y = 2*cos(2x) + 7*cos(x)","instructions":"1. Click the wrench icon (Graph Settings).\n2. Toggle the angle mode from 'Radians' to 'Degrees'.\n3. Set the X-Axis bounds from -180 to 180.\n4. Enter the equation: y = 2*cos(2x) + 7*cos(x)."},"description":"Enter the function and set the viewing window on your graphing tool."}]},{"type":"text","order":2,"title":"Locate the x-intercepts","content":"On the screen, you will observe a symmetric curve that crosses the $x$-axis twice within the window $[-180^\\circ, 180^\\circ]$. \n\nTo find these solutions, use the calculator's tool to locate the **roots** or **zeros** of the function.","calculator":[{"gdc":{"instructions":"1. Analyze Graph -> Zero.\n2. Click to select a lower bound and an upper bound around each intersection point with the x-axis."},"ti84":{"instructions":"1. Press [2nd] [TRACE] (CALC) and select '2: zero'.\n2. Move the cursor to the left of the positive intercept and press [ENTER] (Left Bound).\n3. Move the cursor to the right of the intercept and press [ENTER] (Right Bound).\n4. Press [ENTER] again to guess. \n5. Repeat this process for the negative intercept."},"desmos":{"instructions":"1. Click on the curve.\n2. Click directly on the grey dots representing the x-intercepts to display their coordinates."},"description":"Calculate the exact coordinates of the points where the curve crosses the x-axis."}]},{"type":"text","order":3,"title":"State the Final Solutions","content":"The calculator finds the two $x$-intercepts as:\n\n$$x \\approx 75.522^\\circ \\quad \\text{and} \\quad x \\approx -75.522^\\circ$$\n\nRounding to 1 decimal place as standard for angles in degrees:\n\n$$x \\approx 75.5^\\circ \\quad \\text{and} \\quad x \\approx -75.5^\\circ$$\n\nBoth values lie within the interval $[-180^\\circ, 180^\\circ]$, so they are the valid solutions."}]}],"generatedAt":"2026-06-11T03:45:10.476Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"d9709422-e2ce-44dc-8fb1-5939c3d61419","revisionId":"2c18cf17-f3e9-4af6-9a3c-01e336a53ab0","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"9683a4d6-e50d-4075-aeb8-790ac9572a52","label":"a","content":"Solve, for $0 \\leq x \\leq 360$, the equation,\n\n$$14 \\sin x \\cos x = 5$$\n\nGive your answers to 1 decimal place.","markscheme":"- Uses $2 \\sin x \\cos x = \\sin 2x$ to rewrite as $7 \\sin 2x = 5$ M1\n- $\\sin 2x = \\frac{5}{7}$ M1\n- Finds values for $2x$ in the range $[0, 720]$: $45.58, 134.42, 405.58, 494.42$ M1\n- Divides by 2 to find $x$ M1\n- Final answers: $22.8, 67.2, 202.8, 247.2$ 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 $\\sin 2x = 5/7$.","code":"M"},{"id":"p3","marks":1,"description":"Finds at least two correct values for $2x$ (e.g., $45.58, 134.42$).","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: $22.8, 67.2, 202.8, 247.2$.","code":"A"}],"finalAnswers":["22.8, 67.2, 202.8, 247.2"],"generatedAt":"2026-05-06T14:03:58.962Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"double_angle","label":"Double Angle Identity","steps":[{"type":"text","order":0,"title":"Apply double-angle identity","content":"Recall the double-angle identity for sine from your reference booklet:\n\n$$\\sin 2x = 2 \\sin x \\cos x$$\n\nWe can rewrite the left-hand side of our equation, $14 \\sin x \\cos x$, by factoring out a $7$:\n\n$$14 \\sin x \\cos x = 7(2 \\sin x \\cos x) = 7 \\sin 2x$$\n\nNow, substitute this back into the original equation:\n\n$$7 \\sin 2x = 5$$"},{"type":"text","order":1,"title":"Simplify the equation","content":"Divide both sides of the equation by $7$ to isolate $\\sin 2x$:\n\n$$\\sin 2x = \\frac{5}{7}$$\n\nThis simple form allows us to solve for the angle $2x$ directly."},{"type":"text","order":2,"title":"Adjust the interval","content":"Our original interval for $x$ is:\n\n$$0^{\\circ} \\le x \\le 360^{\\circ}$$\n\nSince we are solving for $2x$, we must adjust the interval by multiplying all parts by $2$:\n\n$$0^{\\circ} \\le 2x \\le 720^{\\circ}$$\n\nWe need to find all solutions for $2x$ within this larger interval."},{"type":"text","order":3,"title":"Find principal values","content":"First, find the principal value by taking the inverse sine:\n\n$$2x = \\arcsin\\left(\\frac{5}{7}\\right) \\approx 45.58^{\\circ}$$\n\nSince sine is positive in the first and second quadrants, the second quadrant solution is:\n\n$$2x = 180^{\\circ} - 45.58^{\\circ} \\approx 134.42^{\\circ}$$"},{"type":"text","order":4,"title":"Find all values of 2x","content":"To find the remaining solutions within the range $0^{\\circ} \\le 2x \\le 720^{\\circ}$, add the period of sine ($360^{\\circ}$) to our first two solutions:\n\n$$2x = 45.58^{\\circ} + 360^{\\circ} = 405.58^{\\circ}$$\n\n$$2x = 134.42^{\\circ} + 360^{\\circ} = 494.42^{\\circ}$$\n\nSo, the four solutions for $2x$ in the interval $[0^{\\circ}, 720^{\\circ}]$ are:\n\n$$2x \\approx 45.58^{\\circ},\\ 134.42^{\\circ},\\ 405.58^{\\circ},\\ 494.42^{\\circ}$$"},{"type":"text","order":5,"title":"Solve for x","content":"Finally, divide each value of $2x$ by $2$ to find the values of $x$:\n\n$$x = \\frac{45.58^{\\circ}}{2} \\approx 22.8^{\\circ}$$\n\n$$x = \\frac{134.42^{\\circ}}{2} \\approx 67.2^{\\circ}$$\n\n$$x = \\frac{405.58^{\\circ}}{2} \\approx 202.8^{\\circ}$$\n\n$$x = \\frac{494.42^{\\circ}}{2} \\approx 247.2^{\\circ}$$\n\nAll of these values lie within the required interval $0^{\\circ} \\le x \\le 360^{\\circ}$.\n\nThus, our final solutions rounded to 1 decimal place are:\n\n$$x = 22.8^{\\circ},\\ 67.2^{\\circ},\\ 202.8^{\\circ},\\ 247.2^{\\circ}$$\n\n**Marks Breakdown:**\n- **M1**: Rewriting the equation using the identity as $7\\sin 2x = 5$.\n- **M1**: Finding $\\sin 2x = \\frac{5}{7}$.\n- **M1**: Finding solutions for $2x$ in the range $[0^{\\circ}, 720^{\\circ}]$.\n- **M1**: Dividing the angles by $2$ to find $x$.\n- **A1**: Correctly rounded final values: $22.8^{\\circ}, 67.2^{\\circ}, 202.8^{\\circ}, 247.2^{\\circ}$."}]},{"id":"tangent_quadratic","label":"Expressing as a Quadratic in Tangent","steps":[{"type":"text","order":0,"title":"Substitute the identity","content":"To solve this equation by expressing it as a quadratic in tangent, we start by replacing the constant $5$ on the right-hand side. \n\nWe know the fundamental trigonometric identity:\n\n$$\\sin^2 x + \\cos^2 x = 1$$\n\nMultiplying this identity by $5$ gives:\n\n$$5 = 5(\\sin^2 x + \\cos^2 x)$$\n\nNow, substitute this expression into our original equation:\n\n$$14 \\sin x \\cos x = 5(\\sin^2 x + \\cos^2 x)$$\n\nExpanding the right-hand side, we get:\n\n$$14 \\sin x \\cos x = 5 \\sin^2 x + 5 \\cos^2 x$$"},{"type":"text","order":1,"title":"Divide by cos^2(x)","content":"To convert the equation into terms of $\\tan x$, we divide every term in the equation by $\\cos^2 x$ (assuming $\\cos x \\neq 0$, which is valid since if $\\cos x = 0$, the equation would lead to a contradiction $0 = 5$):\n\n$$\\frac{14 \\sin x \\cos x}{\\cos^2 x} = \\frac{5 \\sin^2 x}{\\cos^2 x} + \\frac{5 \\cos^2 x}{\\cos^2 x}$$\n\nSimplifying each term using the identity $\\tan x = \\frac{\\sin x}{\\cos x}$:\n\n$$14 \\tan x = 5 \\tan^2 x + 5$$"},{"type":"text","order":2,"title":"Rearrange into a quadratic","content":"Rearrange the equation into the standard quadratic form $a y^2 + b y + c = 0$ where $y = \\tan x$:\n\n$$5 \\tan^2 x - 14 \\tan x + 5 = 0$$"},{"type":"text","order":3,"title":"Solve the quadratic equation","content":"We can solve this quadratic equation using the quadratic formula, where $a = 5$, $b = -14$, and $c = 5$:\n\n$$\\tan x = \\frac{-(-14) \\pm \\sqrt{(-14)^2 - 4(5)(5)}}{2(5)}$$\n\n$$\\tan x = \\frac{14 \\pm \\sqrt{196 - 100}}{10}$$\n\n$$\\tan x = \\frac{14 \\pm \\sqrt{96}}{10}$$\n\nUsing $\\sqrt{96} \\approx 9.79796$, we get two possible values for $\\tan x$:\n\n1. $$\\tan x = \\frac{14 + 9.79796}{10} \\approx 2.3798$$\n2. $$\\tan x = \\frac{14 - 9.79796}{10} \\approx 0.4202$$"},{"type":"text","order":4,"title":"Find the values of x","content":"Now, we solve for $x$ in the interval $0^\\circ \\le x \\le 360^\\circ$ for each case:\n\n**Case 1: $\\tan x \\approx 2.3798$**\n* The principal value is:\n $$x = \\arctan(2.3798) \\approx 67.2^\\circ$$\n* Since tangent has a period of $180^\\circ$, the other solution in the range is:\n $$x = 67.2^\\circ + 180^\\circ = 247.2^\\circ$$\n\n**Case 2: $\\tan x \\approx 0.4202$**\n* The principal value is:\n $$x = \\arctan(0.4202) \\approx 22.8^\\circ$$\n* The other solution in the range is:\n $$x = 22.8^\\circ + 180^\\circ = 202.8^\\circ$$\n\nThus, our final solutions to 1 decimal place are:\n\n$$x = 22.8^\\circ,\\ 67.2^\\circ,\\ 202.8^\\circ,\\ 247.2^\\circ$$"}]}],"generatedAt":"2026-06-09T09:26:01.542Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"9683a4d6-e50d-4075-aeb8-790ac9572a52:double_angle","label":"Double Angle Identity","steps":[{"type":"text","order":0,"title":"Apply double-angle identity","content":"Recall the double-angle identity for sine from your reference booklet:\n\n$$\\sin 2x = 2 \\sin x \\cos x$$\n\nWe can rewrite the left-hand side of our equation, $14 \\sin x \\cos x$, by factoring out a $7$:\n\n$$14 \\sin x \\cos x = 7(2 \\sin x \\cos x) = 7 \\sin 2x$$\n\nNow, substitute this back into the original equation:\n\n$$7 \\sin 2x = 5$$"},{"type":"text","order":1,"title":"Simplify the equation","content":"Divide both sides of the equation by $7$ to isolate $\\sin 2x$:\n\n$$\\sin 2x = \\frac{5}{7}$$\n\nThis simple form allows us to solve for the angle $2x$ directly."},{"type":"text","order":2,"title":"Adjust the interval","content":"Our original interval for $x$ is:\n\n$$0^{\\circ} \\le x \\le 360^{\\circ}$$\n\nSince we are solving for $2x$, we must adjust the interval by multiplying all parts by $2$:\n\n$$0^{\\circ} \\le 2x \\le 720^{\\circ}$$\n\nWe need to find all solutions for $2x$ within this larger interval."},{"type":"text","order":3,"title":"Find principal values","content":"First, find the principal value by taking the inverse sine:\n\n$$2x = \\arcsin\\left(\\frac{5}{7}\\right) \\approx 45.58^{\\circ}$$\n\nSince sine is positive in the first and second quadrants, the second quadrant solution is:\n\n$$2x = 180^{\\circ} - 45.58^{\\circ} \\approx 134.42^{\\circ}$$"},{"type":"text","order":4,"title":"Find all values of 2x","content":"To find the remaining solutions within the range $0^{\\circ} \\le 2x \\le 720^{\\circ}$, add the period of sine ($360^{\\circ}$) to our first two solutions:\n\n$$2x = 45.58^{\\circ} + 360^{\\circ} = 405.58^{\\circ}$$\n\n$$2x = 134.42^{\\circ} + 360^{\\circ} = 494.42^{\\circ}$$\n\nSo, the four solutions for $2x$ in the interval $[0^{\\circ}, 720^{\\circ}]$ are:\n\n$$2x \\approx 45.58^{\\circ},\\ 134.42^{\\circ},\\ 405.58^{\\circ},\\ 494.42^{\\circ}$$"},{"type":"text","order":5,"title":"Solve for x","content":"Finally, divide each value of $2x$ by $2$ to find the values of $x$:\n\n$$x = \\frac{45.58^{\\circ}}{2} \\approx 22.8^{\\circ}$$\n\n$$x = \\frac{134.42^{\\circ}}{2} \\approx 67.2^{\\circ}$$\n\n$$x = \\frac{405.58^{\\circ}}{2} \\approx 202.8^{\\circ}$$\n\n$$x = \\frac{494.42^{\\circ}}{2} \\approx 247.2^{\\circ}$$\n\nAll of these values lie within the required interval $0^{\\circ} \\le x \\le 360^{\\circ}$.\n\nThus, our final solutions rounded to 1 decimal place are:\n\n$$x = 22.8^{\\circ},\\ 67.2^{\\circ},\\ 202.8^{\\circ},\\ 247.2^{\\circ}$$\n\n**Marks Breakdown:**\n- **M1**: Rewriting the equation using the identity as $7\\sin 2x = 5$.\n- **M1**: Finding $\\sin 2x = \\frac{5}{7}$.\n- **M1**: Finding solutions for $2x$ in the range $[0^{\\circ}, 720^{\\circ}]$.\n- **M1**: Dividing the angles by $2$ to find $x$.\n- **A1**: Correctly rounded final values: $22.8^{\\circ}, 67.2^{\\circ}, 202.8^{\\circ}, 247.2^{\\circ}$."}]},{"id":"9683a4d6-e50d-4075-aeb8-790ac9572a52:tangent_quadratic","label":"Expressing as a Quadratic in Tangent","steps":[{"type":"text","order":0,"title":"Substitute the identity","content":"To solve this equation by expressing it as a quadratic in tangent, we start by replacing the constant $5$ on the right-hand side. \n\nWe know the fundamental trigonometric identity:\n\n$$\\sin^2 x + \\cos^2 x = 1$$\n\nMultiplying this identity by $5$ gives:\n\n$$5 = 5(\\sin^2 x + \\cos^2 x)$$\n\nNow, substitute this expression into our original equation:\n\n$$14 \\sin x \\cos x = 5(\\sin^2 x + \\cos^2 x)$$\n\nExpanding the right-hand side, we get:\n\n$$14 \\sin x \\cos x = 5 \\sin^2 x + 5 \\cos^2 x$$"},{"type":"text","order":1,"title":"Divide by cos^2(x)","content":"To convert the equation into terms of $\\tan x$, we divide every term in the equation by $\\cos^2 x$ (assuming $\\cos x \\neq 0$, which is valid since if $\\cos x = 0$, the equation would lead to a contradiction $0 = 5$):\n\n$$\\frac{14 \\sin x \\cos x}{\\cos^2 x} = \\frac{5 \\sin^2 x}{\\cos^2 x} + \\frac{5 \\cos^2 x}{\\cos^2 x}$$\n\nSimplifying each term using the identity $\\tan x = \\frac{\\sin x}{\\cos x}$:\n\n$$14 \\tan x = 5 \\tan^2 x + 5$$"},{"type":"text","order":2,"title":"Rearrange into a quadratic","content":"Rearrange the equation into the standard quadratic form $a y^2 + b y + c = 0$ where $y = \\tan x$:\n\n$$5 \\tan^2 x - 14 \\tan x + 5 = 0$$"},{"type":"text","order":3,"title":"Solve the quadratic equation","content":"We can solve this quadratic equation using the quadratic formula, where $a = 5$, $b = -14$, and $c = 5$:\n\n$$\\tan x = \\frac{-(-14) \\pm \\sqrt{(-14)^2 - 4(5)(5)}}{2(5)}$$\n\n$$\\tan x = \\frac{14 \\pm \\sqrt{196 - 100}}{10}$$\n\n$$\\tan x = \\frac{14 \\pm \\sqrt{96}}{10}$$\n\nUsing $\\sqrt{96} \\approx 9.79796$, we get two possible values for $\\tan x$:\n\n1. $$\\tan x = \\frac{14 + 9.79796}{10} \\approx 2.3798$$\n2. $$\\tan x = \\frac{14 - 9.79796}{10} \\approx 0.4202$$"},{"type":"text","order":4,"title":"Find the values of x","content":"Now, we solve for $x$ in the interval $0^\\circ \\le x \\le 360^\\circ$ for each case:\n\n**Case 1: $\\tan x \\approx 2.3798$**\n* The principal value is:\n $$x = \\arctan(2.3798) \\approx 67.2^\\circ$$\n* Since tangent has a period of $180^\\circ$, the other solution in the range is:\n $$x = 67.2^\\circ + 180^\\circ = 247.2^\\circ$$\n\n**Case 2: $\\tan x \\approx 0.4202$**\n* The principal value is:\n $$x = \\arctan(0.4202) \\approx 22.8^\\circ$$\n* The other solution in the range is:\n $$x = 22.8^\\circ + 180^\\circ = 202.8^\\circ$$\n\nThus, our final solutions to 1 decimal place are:\n\n$$x = 22.8^\\circ,\\ 67.2^\\circ,\\ 202.8^\\circ,\\ 247.2^\\circ$$"}]}],"generatedAt":"2026-06-09T09:26:13.564Z","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 13

7.2 Double Angle Formulae Questions

7.2 Double Angle Formulae

Question 13

7.2 Double Angle Formulae Questions