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bg-background","children":["$","$L3a",null,{"backHref":"/a-level/maths/edexcel/questions","basePath":"/a-level/maths/edexcel/a-level-pure-mathematics-7-2-double-angle-formulae/questions","formulaUrl":"https://assets.mathsgenie.co.uk/board-resources/a-level-maths/edexcel/formula-book.pdf","initialIndex":14,"initialQuestionWindow":[{"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},{"id":"83595e3c-e503-4372-8f45-0c7c60173098","revisionId":"b7eaedff-f322-49e5-b4d2-6c4383f2f8a6","questionNumber":"5","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"7.1 Addition Formulae","calculatorUse":null,"isCalculation":true,"parts":[{"id":"864f4907-d937-4ebc-b2d3-92c49722f31e","label":"0","content":"Solve, for $0 \\leq x \\leq 360$, the equation,\n\n$$8 \\sin x \\cos x = 3$$\n\nGive your answers to 1 decimal place.","markscheme":"- Uses $2 \\sin x \\cos x = \\sin 2x$ to rewrite as $4 \\sin 2x = 3$ M1\n- $\\sin 2x = 0.75$ M1\n- Finds values for $2x$: $48.59, 131.41, 408.59, 491.41$ M1\n- Divides by 2 to find $x$ M1\n- Final answers: $24.3, 65.7, 204.3, 245.7$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Uses the double angle identity to obtain $4 \\sin 2x = 3$.","code":"M"},{"id":"p2","marks":1,"description":"Finds the principal value for $2x \\approx 48.6^\\circ$.","code":"M"},{"id":"p3","marks":1,"description":"Finds at least one other value for $2x$ in the range $[0, 720]$.","code":"M"},{"id":"p4","marks":1,"description":"Divides $2x$ values by 2 to find $x$.","code":"M"},{"id":"p5","marks":1,"description":"Provides all four correct answers: $24.3, 65.7, 204.3, 245.7$.","code":"A"}],"finalAnswers":["24.3","65.7","204.3","245.7"],"generatedAt":"2026-05-06T03:35:10.313Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"double_angle","label":"Double Angle Identity","steps":[{"type":"text","order":0,"title":"Use double angle identity","content":"The given equation is:\n\n$$8 \\sin x \\cos x = 3$$\n\nTo simplify the left-hand side, we recall the double-angle identity for sine:\n\n$$\\sin 2x = 2 \\sin x \\cos x$$\n\nWe can factor out a $4$ from the term on the left-hand side to match the identity:\n\n$$8 \\sin x \\cos x = 4(2 \\sin x \\cos x) = 4 \\sin 2x$$"},{"type":"text","order":1,"title":"Simplify the equation","content":"Substitute the identity back into the original equation:\n\n$$4 \\sin 2x = 3$$\n\nNow divide both sides by $4$ to isolate the trigonometric function:\n\n$$\\sin 2x = 0.75$$"},{"type":"text","order":2,"title":"Adjust the interval","content":"The original range for $x$ is given as:\n\n$$0^\\circ \\leq x \\leq 360^\\circ$$\n\nSince our trigonometric term has the argument $2x$, we must adjust the domain by multiplying all parts of the inequality by $2$:\n\n$$0^\\circ \\leq 2x \\leq 720^\\circ$$"},{"type":"text","order":3,"title":"Find primary solutions","content":"We solve $\\sin 2x = 0.75$ for $2x$ in the interval $[0^\\circ, 720^\\circ]$.\n\nFirst, find the principal value using the inverse sine function:\n\n$$2x = \\arcsin(0.75) \\approx 48.59^\\circ$$\n\nSince the sine function is positive in both the first and second quadrants, the second solution in the first cycle ($0^\\circ$ to $360^\\circ$) is:\n\n$$2x = 180^\\circ - 48.59^\\circ = 131.41^\\circ$$"},{"type":"text","order":4,"title":"Find remaining solutions","content":"Since our interval for $2x$ goes up to $720^\\circ$, we must find the corresponding solutions in the second full rotation. We do this by adding $360^\\circ$ to each of our first two solutions:\n\n$$2x = 48.59^\\circ + 360^\\circ = 408.59^\\circ$$\n\n$$2x = 131.41^\\circ + 360^\\circ = 491.41^\\circ$$\n\nThis gives us four values for $2x$ within our adjusted range:\n\n$$2x \\approx 48.59^\\circ, \\; 131.41^\\circ, \\; 408.59^\\circ, \\; 491.41^\\circ$$"},{"type":"text","order":5,"title":"Solve for x","content":"Finally, divide each value by $2$ to find $x$, and round to 1 decimal place:\n\n$$x_1 = \\frac{48.59^\\circ}{2} \\approx 24.3^\\circ$$\n\n$$x_2 = \\frac{131.41^\\circ}{2} \\approx 65.7^\\circ$$\n\n$$x_3 = \\frac{408.59^\\circ}{2} \\approx 204.3^\\circ$$\n\n$$x_4 = \\frac{491.41^\\circ}{2} \\approx 245.7^\\circ$$\n\nThus, the solutions are $x = 24.3^\\circ$, $65.7^\\circ$, $204.3^\\circ$, and $245.7^\\circ$.","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x), f2(x) = 3","instructions":"1. Set the calculator's angle setting to Degrees.\n2. Open the Graph tool and graph f1(x) = 8*sin(x)*cos(x) and f2(x) = 3.\n3. Set the window dimensions with x from 0 to 360.\n4. Use the Intersection tool to locate all intersection points."},"ti84":{"expression":"Y1 = 8*sin(X)*cos(X)\nY2 = 3","instructions":"1. Change MODE to DEGREE.\n2. Press [Y=] and enter Y1 = 8*sin(X)*cos(X) and Y2 = 3.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Ymin = -5, Ymax = 5.\n4. Press [GRAPH].\n5. Press [2nd] [CALC] and choose '5: intersect' to find the x-values of the intersections."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x);\\ y = 3","instructions":"1. Click the wrench icon, make sure 'Degrees' is selected.\n2. Set the x-axis range to [0, 360].\n3. Type: y = 8sin(x)cos(x)\n4. Type: y = 3\n5. Click on the intersection points to find their coordinate values."},"description":"Verify your solutions visually by graphing the original equation."}]}]},{"id":"graphical","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Set up the functions","content":"To solve the equation $8 \\sin x \\cos x = 3$ graphically, we can define two separate functions and find where their graphs intersect on the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n$$y_1 = 8 \\sin x \\cos x$$\n$$y_2 = 3$$"},{"type":"text","order":1,"title":"Configure the calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode since the domain is given in degrees ($0^\\circ \\leq x \\leq 360^\\circ$). Adjust your window settings to view the full domain:\n- **Xmin** $= 0$\n- **Xmax** $= 360$\n- **Ymin** $= -5$\n- **Ymax** $= 5$","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x)\nf2(x) = 3","instructions":"1. Go to the Graph mode.\n2. Set the angle unit to Degree in the settings.\n3. Input the functions:\n \$f1(x) = 8 \\sin(x) \\cos(x)\$\n \$f2(x) = 3\$\n4. Set the View Window: X from 0 to 360, Y from -5 to 5.\n5. Draw the graph."},"ti84":{"expression":"Y_1 = 8*sin(X)*cos(X)\nY_2 = 3","instructions":"1. Press [MODE] and ensure DEGREE is highlighted.\n2. Press [Y=] and enter:\n \$Y_1 = 8 \\sin(X) \\cos(X)\$\n \$Y_2 = 3\$\n3. Press [WINDOW] and set:\n Xmin = 0, Xmax = 360, Xscl = 30\n Ymin = -5, Ymax = 5, Yscl = 1\n4. Press [GRAPH] to display the functions."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x)\ny = 3","instructions":"1. Click on the wrench icon (Graph Settings) and select Degrees.\n2. Set the X-axis limits to [0, 360] and Y-axis limits to [-5, 5].\n3. Type: `y = 8*sin(x)*cos(x)`\n4. Type: `y = 3`"},"description":"Graph the equations and adjust the window settings."}]},{"type":"text","order":2,"title":"Find the intersections","content":"Use your graphing calculator's intersection tool to locate the coordinates of the points where the two curves meet. Within the range $[0, 360]$, you will see four intersection points.","calculator":[{"gdc":{"instructions":"1. Use the G-Solv (Graph Solve) tool.\n2. Select ISCT (Intersection).\n3. Use the arrow keys to cycle through and record all four intersection values."},"ti84":{"instructions":"1. Press [2nd] [CALC] and select option 5: intersect.\n2. Move the cursor near the first intersection point, then press [ENTER] three times. Write down the x-value.\n3. Repeat this process for the other three intersection points by moving the cursor close to each successive peak."},"desmos":{"instructions":"1. Click on the gray intersection points on the graph where the wave meets the line \$y = 3\$.\n2. Note down the x-coordinates displayed."},"description":"Find each of the 4 intersection points on the graph."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The calculator returns the following $x$-coordinates for the intersection points:\n\n* $x \\approx 24.295^\\circ$\n* $x \\approx 65.705^\\circ$\n* $x \\approx 204.295^\\circ$\n* $x \\approx 245.705^\\circ$\n\nRounding each answer to 1 decimal place as required, we obtain:\n\n$$x = 24.3^\\circ, \\; 65.7^\\circ, \\; 204.3^\\circ, \\; 245.7^\\circ$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]}],"generatedAt":"2026-06-09T07:08:37.695Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"864f4907-d937-4ebc-b2d3-92c49722f31e:double_angle","label":"Double Angle Identity","steps":[{"type":"text","order":0,"title":"Use double angle identity","content":"The given equation is:\n\n$$8 \\sin x \\cos x = 3$$\n\nTo simplify the left-hand side, we recall the double-angle identity for sine:\n\n$$\\sin 2x = 2 \\sin x \\cos x$$\n\nWe can factor out a $4$ from the term on the left-hand side to match the identity:\n\n$$8 \\sin x \\cos x = 4(2 \\sin x \\cos x) = 4 \\sin 2x$$"},{"type":"text","order":1,"title":"Simplify the equation","content":"Substitute the identity back into the original equation:\n\n$$4 \\sin 2x = 3$$\n\nNow divide both sides by $4$ to isolate the trigonometric function:\n\n$$\\sin 2x = 0.75$$"},{"type":"text","order":2,"title":"Adjust the interval","content":"The original range for $x$ is given as:\n\n$$0^\\circ \\leq x \\leq 360^\\circ$$\n\nSince our trigonometric term has the argument $2x$, we must adjust the domain by multiplying all parts of the inequality by $2$:\n\n$$0^\\circ \\leq 2x \\leq 720^\\circ$$"},{"type":"text","order":3,"title":"Find primary solutions","content":"We solve $\\sin 2x = 0.75$ for $2x$ in the interval $[0^\\circ, 720^\\circ]$.\n\nFirst, find the principal value using the inverse sine function:\n\n$$2x = \\arcsin(0.75) \\approx 48.59^\\circ$$\n\nSince the sine function is positive in both the first and second quadrants, the second solution in the first cycle ($0^\\circ$ to $360^\\circ$) is:\n\n$$2x = 180^\\circ - 48.59^\\circ = 131.41^\\circ$$"},{"type":"text","order":4,"title":"Find remaining solutions","content":"Since our interval for $2x$ goes up to $720^\\circ$, we must find the corresponding solutions in the second full rotation. We do this by adding $360^\\circ$ to each of our first two solutions:\n\n$$2x = 48.59^\\circ + 360^\\circ = 408.59^\\circ$$\n\n$$2x = 131.41^\\circ + 360^\\circ = 491.41^\\circ$$\n\nThis gives us four values for $2x$ within our adjusted range:\n\n$$2x \\approx 48.59^\\circ, \\; 131.41^\\circ, \\; 408.59^\\circ, \\; 491.41^\\circ$$"},{"type":"text","order":5,"title":"Solve for x","content":"Finally, divide each value by $2$ to find $x$, and round to 1 decimal place:\n\n$$x_1 = \\frac{48.59^\\circ}{2} \\approx 24.3^\\circ$$\n\n$$x_2 = \\frac{131.41^\\circ}{2} \\approx 65.7^\\circ$$\n\n$$x_3 = \\frac{408.59^\\circ}{2} \\approx 204.3^\\circ$$\n\n$$x_4 = \\frac{491.41^\\circ}{2} \\approx 245.7^\\circ$$\n\nThus, the solutions are $x = 24.3^\\circ$, $65.7^\\circ$, $204.3^\\circ$, and $245.7^\\circ$.","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x), f2(x) = 3","instructions":"1. Set the calculator's angle setting to Degrees.\n2. Open the Graph tool and graph f1(x) = 8*sin(x)*cos(x) and f2(x) = 3.\n3. Set the window dimensions with x from 0 to 360.\n4. Use the Intersection tool to locate all intersection points."},"ti84":{"expression":"Y1 = 8*sin(X)*cos(X)\nY2 = 3","instructions":"1. Change MODE to DEGREE.\n2. Press [Y=] and enter Y1 = 8*sin(X)*cos(X) and Y2 = 3.\n3. Press [WINDOW] and set Xmin = 0, Xmax = 360, Ymin = -5, Ymax = 5.\n4. Press [GRAPH].\n5. Press [2nd] [CALC] and choose '5: intersect' to find the x-values of the intersections."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x);\\ y = 3","instructions":"1. Click the wrench icon, make sure 'Degrees' is selected.\n2. Set the x-axis range to [0, 360].\n3. Type: y = 8sin(x)cos(x)\n4. Type: y = 3\n5. Click on the intersection points to find their coordinate values."},"description":"Verify your solutions visually by graphing the original equation."}]}]},{"id":"864f4907-d937-4ebc-b2d3-92c49722f31e:graphical","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Set up the functions","content":"To solve the equation $8 \\sin x \\cos x = 3$ graphically, we can define two separate functions and find where their graphs intersect on the interval $0^\\circ \\leq x \\leq 360^\\circ$:\n\n$$y_1 = 8 \\sin x \\cos x$$\n$$y_2 = 3$$"},{"type":"text","order":1,"title":"Configure the calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode since the domain is given in degrees ($0^\\circ \\leq x \\leq 360^\\circ$). Adjust your window settings to view the full domain:\n- **Xmin** $= 0$\n- **Xmax** $= 360$\n- **Ymin** $= -5$\n- **Ymax** $= 5$","calculator":[{"gdc":{"expression":"f1(x) = 8*sin(x)*cos(x)\nf2(x) = 3","instructions":"1. Go to the Graph mode.\n2. Set the angle unit to Degree in the settings.\n3. Input the functions:\n \$f1(x) = 8 \\sin(x) \\cos(x)\$\n \$f2(x) = 3\$\n4. Set the View Window: X from 0 to 360, Y from -5 to 5.\n5. Draw the graph."},"ti84":{"expression":"Y_1 = 8*sin(X)*cos(X)\nY_2 = 3","instructions":"1. Press [MODE] and ensure DEGREE is highlighted.\n2. Press [Y=] and enter:\n \$Y_1 = 8 \\sin(X) \\cos(X)\$\n \$Y_2 = 3\$\n3. Press [WINDOW] and set:\n Xmin = 0, Xmax = 360, Xscl = 30\n Ymin = -5, Ymax = 5, Yscl = 1\n4. Press [GRAPH] to display the functions."},"desmos":{"expression":"y = 8\\sin(x)\\cos(x)\ny = 3","instructions":"1. Click on the wrench icon (Graph Settings) and select Degrees.\n2. Set the X-axis limits to [0, 360] and Y-axis limits to [-5, 5].\n3. Type: `y = 8*sin(x)*cos(x)`\n4. Type: `y = 3`"},"description":"Graph the equations and adjust the window settings."}]},{"type":"text","order":2,"title":"Find the intersections","content":"Use your graphing calculator's intersection tool to locate the coordinates of the points where the two curves meet. Within the range $[0, 360]$, you will see four intersection points.","calculator":[{"gdc":{"instructions":"1. Use the G-Solv (Graph Solve) tool.\n2. Select ISCT (Intersection).\n3. Use the arrow keys to cycle through and record all four intersection values."},"ti84":{"instructions":"1. Press [2nd] [CALC] and select option 5: intersect.\n2. Move the cursor near the first intersection point, then press [ENTER] three times. Write down the x-value.\n3. Repeat this process for the other three intersection points by moving the cursor close to each successive peak."},"desmos":{"instructions":"1. Click on the gray intersection points on the graph where the wave meets the line \$y = 3\$.\n2. Note down the x-coordinates displayed."},"description":"Find each of the 4 intersection points on the graph."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The calculator returns the following $x$-coordinates for the intersection points:\n\n* $x \\approx 24.295^\\circ$\n* $x \\approx 65.705^\\circ$\n* $x \\approx 204.295^\\circ$\n* $x \\approx 245.705^\\circ$\n\nRounding each answer to 1 decimal place as required, we obtain:\n\n$$x = 24.3^\\circ, \\; 65.7^\\circ, \\; 204.3^\\circ, \\; 245.7^\\circ$$","highlights":[{"text":"Give your answers to 1 decimal place.","location":"specification"}]}]}],"generatedAt":"2026-06-09T07:08:50.985Z","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 14

7.2 Double Angle Formulae Questions

7.2 Double Angle Formulae

Question 14

7.2 Double Angle Formulae Questions