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bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/ccea/questions","basePath":"/a-level/maths/ccea/trigonometry-7/questions","formulaUrl":"https://ccea.org.uk/downloads/docs/Support/General/2019/Mathematical%20Formulae%20and%20Tables.pdf","initialIndex":2,"initialQuestionWindow":[{"id":"4ed7094e-cf03-4e1e-8691-1d3d0175a970","revisionId":"de27527b-7861-4ecc-b905-480729bd71eb","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"2dd0e67c-5b44-42a8-b80f-3e556de903a9","label":"a","content":"Solve, for $-180 \\leq x \\leq 180$, the equation,\n\n$$\\cos 2x + 3 \\cos x - 1 = 0$$","markscheme":"- Substitutes $\\cos 2x = 2 \\cos^2 x - 1$ M1\n- Forms quadratic: $2 \\cos^2 x + 3 \\cos x - 2 = 0$ M1\n- Factorises to $(2 \\cos x - 1)(\\cos x + 2) = 0$ M1\n- Solves $\\cos x = \\frac{1}{2}$ (rejects $\\cos x = -2$) M1\n- Final answers: $x = 60^\\circ, -60^\\circ$ 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 $\\cos x = 1/2$ and explicit or implied rejection of the other root.","code":"M"},{"id":"p5","marks":1,"description":"Final answers $x = 60^\\circ$ and $x = -60^\\circ$ provided correctly in range.","code":"A"}],"finalAnswers":["60^\\circ, -60^\\circ"],"generatedAt":"2026-05-06T14:04:20.788Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"algebraic","label":"Algebraic Substitution","steps":[{"type":"text","order":0,"title":"Use double-angle identity","content":"To solve the equation $\\cos 2x + 3 \\cos x - 1 = 0$, we first need to express all trigonometric terms in terms of a single angle, $x$.\n\nWe can use the double-angle identity for cosine from our formula sheet:\n$$\\cos 2x = 2 \\cos^2 x - 1$$\n\nSubstituting this into our equation gives:\n$$(2 \\cos^2 x - 1) + 3 \\cos x - 1 = 0$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"Next, we simplify the terms to form a standard quadratic equation in terms of $\\cos x$:\n\n$$2 \\cos^2 x + 3 \\cos x - 2 = 0$$\n\nTo make this easier to solve, we can use algebraic substitution. Let $y = \\cos x$. This transforms our equation into a standard quadratic equation:\n\n$$2y^2 + 3y - 2 = 0$$"},{"type":"text","order":2,"title":"Factorise the quadratic","content":"Now we factorise the quadratic equation $2y^2 + 3y - 2 = 0$.\n\nWe look for two numbers that multiply to $2 \\times (-2) = -4$ and add to $3$. These numbers are $4$ and $-1$.\n\nRewriting and factorising by grouping:\n$$\\begin{aligned} 2y^2 + 4y - y - 2 &= 0 \\\\ 2y(y + 2) - 1(y + 2) &= 0 \\\\ (2y - 1)(y + 2) &= 0 \\end{aligned}$$\n\nSubstituting $\\cos x$ back in for $y$, we get:\n$$(2 \\cos x - 1)(\\cos x + 2) = 0$$"},{"type":"text","order":3,"title":"Solve for cos x","content":"Setting each factor to zero gives two possible branches:\n\n1. $$2 \\cos x - 1 = 0 \\implies \\cos x = \\frac{1}{2}$$\n2. $$\\cos x + 2 = 0 \\implies \\cos x = -2$$\n\nSince the range of the cosine function is $-1 \\leq \\cos x \\leq 1$, the equation $\\cos x = -2$ has no real solutions. Thus, we reject this branch and focus solely on:\n$$\\cos x = \\frac{1}{2}$$"},{"type":"text","order":4,"title":"Find the final angles","content":"We now find the values of $x$ in the interval $-180^\\circ \\leq x \\leq 180^\\circ$ that satisfy $\\cos x = \\frac{1}{2}$.\n\nThe principal value is:\n$$x = \\arccos\\left(\\frac{1}{2}\\right) = 60^\\circ$$\n\nUsing the symmetry of the cosine graph, $\\cos(-x) = \\cos(x)$, the second solution within our domain is:\n$$x = -60^\\circ$$\n\nBoth values lie within the required interval $[-180^\\circ, 180^\\circ]$.\n\nTherefore, the final solutions are:\n$$x = 60^\\circ, -60^\\circ$$"}]},{"id":"graphical","label":"Graphical Approach","steps":[{"type":"text","order":0,"title":"Define the function","content":"To solve the equation using a graphical approach, we define a function $y = f(x)$ representing the left side of the equation:\n\n$$y = \\cos(2x) + 3\\cos(x) - 1$$\n\nWe need to find the $x$-intercepts (roots) of this function where $y = 0$, within the specified interval:\n\n$$-180^\\circ ≤ x ≤ 180^\\circ$$"},{"type":"text","order":1,"title":"Set up your calculator","content":"Since the interval is specified in degrees (from $-180^\\circ$ to $180^\\circ$), make sure your graphing calculator is set to **Degree mode** before entering the function and adjusting the window settings.","calculator":[{"gdc":{"expression":"cos(2x) + 3cos(x) - 1","instructions":"1. Go to Graph mode and set Angle to 'Degree'.\n2. Enter the function: y1 = cos(2x) + 3cos(x) - 1.\n3. Set the View-Window (V-Window) for x to [-180, 180] and y to [-5, 5].\n4. Draw the graph."},"ti84":{"expression":"cos(2X) + 3cos(X) - 1","instructions":"1. Press [MODE] and ensure 'DEGREE' is highlighted.\n2. Press [Y=] and enter Y1 = cos(2X) + 3cos(X) - 1.\n3. Press [WINDOW] and set:\n - Xmin = -180\n - Xmax = 180\n - Xscl = 30\n - Ymin = -5\n - Ymax = 5\n4. Press [GRAPH]."},"desmos":{"expression":"y = \\cos(2x) + 3\\cos(x) - 1","instructions":"1. Click on the Graph Settings wrench icon in the top right corner.\n2. Select 'Degrees'.\n3. Type in the equation: y = cos(2x) + 3cos(x) - 1.\n4. Set the X-axis bounds to [-180, 180] and the Y-axis bounds to [-5, 5]."},"description":"Configure your calculator to graph the function $y = \\cos(2x) + 3\\cos(x) - 1$ in degree mode."}]},{"type":"text","order":2,"title":"Locate the x-intercepts","content":"Looking at the generated graph, you will see a symmetric curve with two $x$-intercepts within the interval $[-180^\\circ, 180^\\circ]$.\n\nWe can find these precise points using the zero/root-finding feature of the graphing tool:\n\n- The first intercept on the negative side is located at:\n $$x = -60^\\circ$$\n\n- The second intercept on the positive side is located at:\n $$x = 60^\\circ$$","calculator":[{"gdc":{"instructions":"1. Press [G-Solve] (usually [F5]).\n2. Select [Root] (usually [F1]).\n3. Use the arrow keys to toggle between the two solutions: x = -60° and x = 60°."},"ti84":{"instructions":"1. Press [2nd] then [CALC] (above [TRACE]) and select '2: zero'.\n2. For the left root: Set a Left Bound to the left of -60 (e.g., -90), a Right Bound to the right (e.g., -30), and press [ENTER] to guess. The calculator displays X = -60.\n3. Repeat for the right root: Set a Left Bound (e.g., 30), a Right Bound (e.g., 90), and press [ENTER] to guess. The calculator displays X = 60."},"desmos":{"instructions":"1. Simply click on the points where the curve crosses the x-axis.\n2. Desmos will highlight the coordinates as grey dots, displaying (-60, 0) and (60, 0)."},"description":"Find the roots of the equation on your graphing tool."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The values of $x$ in the interval $[-180^\\circ, 180^\\circ]$ that satisfy the equation are:\n\n$$x = -60^\\circ \\quad \\text{and} \\quad x = 60^\\circ$$"}]}],"generatedAt":"2026-06-09T05:35:26.170Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"2dd0e67c-5b44-42a8-b80f-3e556de903a9:algebraic","label":"Algebraic Substitution","steps":[{"type":"text","order":0,"title":"Use double-angle identity","content":"To solve the equation $\\cos 2x + 3 \\cos x - 1 = 0$, we first need to express all trigonometric terms in terms of a single angle, $x$.\n\nWe can use the double-angle identity for cosine from our formula sheet:\n$$\\cos 2x = 2 \\cos^2 x - 1$$\n\nSubstituting this into our equation gives:\n$$(2 \\cos^2 x - 1) + 3 \\cos x - 1 = 0$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"Next, we simplify the terms to form a standard quadratic equation in terms of $\\cos x$:\n\n$$2 \\cos^2 x + 3 \\cos x - 2 = 0$$\n\nTo make this easier to solve, we can use algebraic substitution. Let $y = \\cos x$. This transforms our equation into a standard quadratic equation:\n\n$$2y^2 + 3y - 2 = 0$$"},{"type":"text","order":2,"title":"Factorise the quadratic","content":"Now we factorise the quadratic equation $2y^2 + 3y - 2 = 0$.\n\nWe look for two numbers that multiply to $2 \\times (-2) = -4$ and add to $3$. These numbers are $4$ and $-1$.\n\nRewriting and factorising by grouping:\n$$\\begin{aligned} 2y^2 + 4y - y - 2 &= 0 \\\\ 2y(y + 2) - 1(y + 2) &= 0 \\\\ (2y - 1)(y + 2) &= 0 \\end{aligned}$$\n\nSubstituting $\\cos x$ back in for $y$, we get:\n$$(2 \\cos x - 1)(\\cos x + 2) = 0$$"},{"type":"text","order":3,"title":"Solve for cos x","content":"Setting each factor to zero gives two possible branches:\n\n1. $$2 \\cos x - 1 = 0 \\implies \\cos x = \\frac{1}{2}$$\n2. $$\\cos x + 2 = 0 \\implies \\cos x = -2$$\n\nSince the range of the cosine function is $-1 \\leq \\cos x \\leq 1$, the equation $\\cos x = -2$ has no real solutions. Thus, we reject this branch and focus solely on:\n$$\\cos x = \\frac{1}{2}$$"},{"type":"text","order":4,"title":"Find the final angles","content":"We now find the values of $x$ in the interval $-180^\\circ \\leq x \\leq 180^\\circ$ that satisfy $\\cos x = \\frac{1}{2}$.\n\nThe principal value is:\n$$x = \\arccos\\left(\\frac{1}{2}\\right) = 60^\\circ$$\n\nUsing the symmetry of the cosine graph, $\\cos(-x) = \\cos(x)$, the second solution within our domain is:\n$$x = -60^\\circ$$\n\nBoth values lie within the required interval $[-180^\\circ, 180^\\circ]$.\n\nTherefore, the final solutions are:\n$$x = 60^\\circ, -60^\\circ$$"}]},{"id":"2dd0e67c-5b44-42a8-b80f-3e556de903a9:graphical","label":"Graphical Approach","steps":[{"type":"text","order":0,"title":"Define the function","content":"To solve the equation using a graphical approach, we define a function $y = f(x)$ representing the left side of the equation:\n\n$$y = \\cos(2x) + 3\\cos(x) - 1$$\n\nWe need to find the $x$-intercepts (roots) of this function where $y = 0$, within the specified interval:\n\n$$-180^\\circ ≤ x ≤ 180^\\circ$$"},{"type":"text","order":1,"title":"Set up your calculator","content":"Since the interval is specified in degrees (from $-180^\\circ$ to $180^\\circ$), make sure your graphing calculator is set to **Degree mode** before entering the function and adjusting the window settings.","calculator":[{"gdc":{"expression":"cos(2x) + 3cos(x) - 1","instructions":"1. Go to Graph mode and set Angle to 'Degree'.\n2. Enter the function: y1 = cos(2x) + 3cos(x) - 1.\n3. Set the View-Window (V-Window) for x to [-180, 180] and y to [-5, 5].\n4. Draw the graph."},"ti84":{"expression":"cos(2X) + 3cos(X) - 1","instructions":"1. Press [MODE] and ensure 'DEGREE' is highlighted.\n2. Press [Y=] and enter Y1 = cos(2X) + 3cos(X) - 1.\n3. Press [WINDOW] and set:\n - Xmin = -180\n - Xmax = 180\n - Xscl = 30\n - Ymin = -5\n - Ymax = 5\n4. Press [GRAPH]."},"desmos":{"expression":"y = \\cos(2x) + 3\\cos(x) - 1","instructions":"1. Click on the Graph Settings wrench icon in the top right corner.\n2. Select 'Degrees'.\n3. Type in the equation: y = cos(2x) + 3cos(x) - 1.\n4. Set the X-axis bounds to [-180, 180] and the Y-axis bounds to [-5, 5]."},"description":"Configure your calculator to graph the function $y = \\cos(2x) + 3\\cos(x) - 1$ in degree mode."}]},{"type":"text","order":2,"title":"Locate the x-intercepts","content":"Looking at the generated graph, you will see a symmetric curve with two $x$-intercepts within the interval $[-180^\\circ, 180^\\circ]$.\n\nWe can find these precise points using the zero/root-finding feature of the graphing tool:\n\n- The first intercept on the negative side is located at:\n $$x = -60^\\circ$$\n\n- The second intercept on the positive side is located at:\n $$x = 60^\\circ$$","calculator":[{"gdc":{"instructions":"1. Press [G-Solve] (usually [F5]).\n2. Select [Root] (usually [F1]).\n3. Use the arrow keys to toggle between the two solutions: x = -60° and x = 60°."},"ti84":{"instructions":"1. Press [2nd] then [CALC] (above [TRACE]) and select '2: zero'.\n2. For the left root: Set a Left Bound to the left of -60 (e.g., -90), a Right Bound to the right (e.g., -30), and press [ENTER] to guess. The calculator displays X = -60.\n3. Repeat for the right root: Set a Left Bound (e.g., 30), a Right Bound (e.g., 90), and press [ENTER] to guess. The calculator displays X = 60."},"desmos":{"instructions":"1. Simply click on the points where the curve crosses the x-axis.\n2. Desmos will highlight the coordinates as grey dots, displaying (-60, 0) and (60, 0)."},"description":"Find the roots of the equation on your graphing tool."}]},{"type":"text","order":3,"title":"State the final solutions","content":"The values of $x$ in the interval $[-180^\\circ, 180^\\circ]$ that satisfy the equation are:\n\n$$x = -60^\\circ \\quad \\text{and} \\quad x = 60^\\circ$$"}]}],"generatedAt":"2026-06-09T05:35:34.356Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"2cb328b9-19c8-4983-a09d-31ba9ff0ece8","revisionId":"1ce25589-1c08-4e2e-84f7-053990f1bd7f","questionNumber":"4","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":"7.1 Addition Formulae","calculatorUse":null,"isCalculation":true,"parts":[{"id":"9e48ba86-fda2-4e28-9006-ef63a34c5114","label":"0","content":"Solve, for $-180 \\leq x \\leq 180$, the equation,\n\n$$\\cos 2x - 7 \\sin x + 3 = 0$$","markscheme":"- Substitutes $\\cos 2x = 1 - 2 \\sin^2 x$ M1\n- Forms quadratic: $2 \\sin^2 x + 7 \\sin x - 4 = 0$ M1\n- Factorises to $(2 \\sin x - 1)(\\sin x + 4) = 0$ M1\n- Solves $\\sin x = \\frac{1}{2}$ (rejects $\\sin x = -4$) M1\n- Final answers: $x = 30^\\circ, 150^\\circ$ (or $\\frac{1}{6}\\pi, \\frac{5}{6}\\pi$) A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Uses $\\cos 2x = 1 - 2 \\sin^2 x$ to create an equation in $\\sin x$.","code":"M"},{"id":"p2","marks":1,"description":"Rearranges to a standard quadratic form $2 \\sin^2 x + 7 \\sin x - 4 = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Factorises or uses quadratic formula to find $\\sin x = 0.5$.","code":"M"},{"id":"p4","marks":1,"description":"Finds the principal value $x = 30^\\circ$.","code":"M"},{"id":"p5","marks":1,"description":"Finds the second value $x = 150^\\circ$ and no others in range.","code":"A"}],"finalAnswers":["30^\\circ","150^\\circ"],"generatedAt":"2026-05-06T03:35:10.313Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"algebraic","label":"Algebraic with Double-Angle Identity","steps":[{"type":"text","order":0,"title":"Use double-angle identity","content":"To solve the equation, we first want to rewrite it so that it only contains a single trigonometric function. Since there is a $-7 \\sin x$ term, we should express $\\cos 2x$ in terms of $\\sin x$.\n\nFrom the double-angle identity for cosine:\n\n$$\\cos 2x = 1 - 2\\sin^2 x$$\n\nSubstituting this into our equation gives:\n\n$$(1 - 2\\sin^2 x) - 7\\sin x + 3 = 0$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"Now, we simplify and rearrange the terms to form a standard quadratic equation in terms of $\\sin x$.\n\nCombining the constant terms:\n\n$$-2\\sin^2 x - 7\\sin x + 4 = 0$$\n\nMultiply the entire equation by $-1$ to make the leading coefficient positive, making it easier to factorise:\n\n$$2\\sin^2 x + 7\\sin x - 4 = 0$$"},{"type":"text","order":2,"title":"Factorise the quadratic","content":"Let's factorise the quadratic equation $2\\sin^2 x + 7\\sin x - 4 = 0$.\n\nWe need two numbers that multiply to $2 \\times (-4) = -8$ and add to $7$. These numbers are $8$ and $-1$. \n\nSplitting the middle term:\n\n$$2\\sin^2 x + 8\\sin x - \\sin x - 4 = 0$$\n\n$$2\\sin x(\\sin x + 4) - 1(\\sin x + 4) = 0$$\n\n$$(2\\sin x - 1)(\\sin x + 4) = 0$$"},{"type":"text","order":3,"title":"Solve for sin x","content":"Now, we set each factor equal to zero to find the possible values for $\\sin x$:\n\n1. **First factor:**\n $$2\\sin x - 1 = 0 \\implies \\sin x = \\frac{1}{2}$$\n\n2. **Second factor:**\n $$\\sin x + 4 = 0 \\implies \\sin x = -4$$\n\nSince the value of the sine function must lie between $-1$ and $1$ (inclusive), the equation $\\sin x = -4$ has **no real solutions**. \n\nThus, we only need to solve:\n\n$$\\sin x = \\frac{1}{2}$$\n\nfor $-180^\\circ \\le x \\le 180^\\circ$."},{"type":"text","order":4,"title":"Find the final solutions","content":"We solve $\\sin x = \\frac{1}{2}$ within the interval $[-180^\\circ, 180^\\circ]$.\n\n* **Principal Value:** \n $$x = \\arcsin\\left(\\frac{1}{2}\\right) = 30^\\circ$$\n\n* **Second Value:** \n Using the symmetry of the sine graph ($180^\\circ - x$):\n $$x = 180^\\circ - 30^\\circ = 150^\\circ$$\n\nBoth $30^\\circ$ and $150^\\circ$ lie within the specified range $[-180^\\circ, 180^\\circ]$.\n\nTherefore, the solutions are:\n\n$$x = 30^\\circ, 150^\\circ$$"}]},{"id":"graphical","label":"Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the function","content":"To solve the equation $\\cos 2x - 7 \\sin x + 3 = 0$ graphically, we can define the left-hand side of the equation as a function of $x$:\n\n$$y = \\cos 2x - 7 \\sin x + 3$$\n\nOur goal is to find the values of $x$ in the interval $[-180^\\circ, 180^\\circ]$ where this function is equal to zero (the $x$-intercepts of the graph)."},{"type":"text","order":1,"title":"Configure the calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode since the interval is given in degrees ($-180^\\circ \\leq x \\leq 180^\\circ$). Adjust the window settings to display the correct interval for $x$ and a suitable range for $y$ to clearly see the intercepts:\n\n- **Xmin:** $-180$\n- **Xmax:** $180$\n- **Xscl:** $30$\n- **Ymin:** $-5$\n- **Ymax:** $15$","calculator":[{"gdc":{"expression":"f1(x) = cos(2x) - 7sin(x) + 3","instructions":"1. Change the angle mode to Degrees in Settings.\n2. Open the Graph app and enter f1(x) = cos(2x) - 7sin(x) + 3.\n3. Adjust the window settings to: X: [-180, 180] and Y: [-5, 15].\n4. Plot the graph."},"ti84":{"expression":"Y1 = cos(2X) - 7sin(X) + 3","instructions":"1. Press [MODE], highlight 'DEGREE', and press [ENTER].\n2. Press [Y=] and enter Y1 = cos(2X) - 7sin(X) + 3.\n3. Press [WINDOW] and set: Xmin = -180, Xmax = 180, Xscl = 30, Ymin = -5, Ymax = 15.\n4. Press [GRAPH] to plot."},"desmos":{"expression":"y = \\cos(2x) - 7\\sin(x) + 3","instructions":"1. Click on the wrench icon (Graph Settings) in the top-right corner.\n2. Select 'Degrees'.\n3. Set the X-axis limits to -180 to 180, and Y-axis limits to -5 to 15.\n4. Type the equation in the input bar."},"description":"Plot the function and set the viewing window in degree mode."}]},{"type":"text","order":2,"title":"Find the intercepts","content":"Look at the plotted graph. Notice that the curve intersects the $x$-axis at two points in the given domain $[-180, 180]$. Use the zero-finding feature of your graphing calculator to find these exact coordinates.","calculator":[{"gdc":{"instructions":"1. Go to Menu -> Analyze Graph -> Zero.\n2. Click to the left and right of each of the two x-intercepts to display their coordinates."},"ti84":{"instructions":"1. Press [2nd] [TRACE] (CALC) and select option 2: 'zero'.\n2. For the first zero, set the Left Bound to the left of the first intercept (e.g., x = 0), the Right Bound to its right (e.g., x = 50), and press [ENTER] on Guess. Repeat for the second intercept with bounds around x = 150.\n3. The calculator displays the solutions."},"desmos":{"instructions":"1. Click directly on the curve.\n2. Click on the grey dots that appear at the x-intercepts to reveal their coordinates."},"description":"Locate the x-intercepts of the curve."}]},{"type":"text","order":3,"title":"State final solutions","content":"The calculator yields the following $x$-intercepts within the interval $[-180^\\circ, 180^\\circ]$:\n\n$$x = 30^\\circ, \\quad x = 150^\\circ$$\n\nThese correspond to the exact solutions of our trigonometric equation."}]}],"generatedAt":"2026-06-09T04:09:04.631Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"9e48ba86-fda2-4e28-9006-ef63a34c5114:algebraic","label":"Algebraic with Double-Angle Identity","steps":[{"type":"text","order":0,"title":"Use double-angle identity","content":"To solve the equation, we first want to rewrite it so that it only contains a single trigonometric function. Since there is a $-7 \\sin x$ term, we should express $\\cos 2x$ in terms of $\\sin x$.\n\nFrom the double-angle identity for cosine:\n\n$$\\cos 2x = 1 - 2\\sin^2 x$$\n\nSubstituting this into our equation gives:\n\n$$(1 - 2\\sin^2 x) - 7\\sin x + 3 = 0$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"Now, we simplify and rearrange the terms to form a standard quadratic equation in terms of $\\sin x$.\n\nCombining the constant terms:\n\n$$-2\\sin^2 x - 7\\sin x + 4 = 0$$\n\nMultiply the entire equation by $-1$ to make the leading coefficient positive, making it easier to factorise:\n\n$$2\\sin^2 x + 7\\sin x - 4 = 0$$"},{"type":"text","order":2,"title":"Factorise the quadratic","content":"Let's factorise the quadratic equation $2\\sin^2 x + 7\\sin x - 4 = 0$.\n\nWe need two numbers that multiply to $2 \\times (-4) = -8$ and add to $7$. These numbers are $8$ and $-1$. \n\nSplitting the middle term:\n\n$$2\\sin^2 x + 8\\sin x - \\sin x - 4 = 0$$\n\n$$2\\sin x(\\sin x + 4) - 1(\\sin x + 4) = 0$$\n\n$$(2\\sin x - 1)(\\sin x + 4) = 0$$"},{"type":"text","order":3,"title":"Solve for sin x","content":"Now, we set each factor equal to zero to find the possible values for $\\sin x$:\n\n1. **First factor:**\n $$2\\sin x - 1 = 0 \\implies \\sin x = \\frac{1}{2}$$\n\n2. **Second factor:**\n $$\\sin x + 4 = 0 \\implies \\sin x = -4$$\n\nSince the value of the sine function must lie between $-1$ and $1$ (inclusive), the equation $\\sin x = -4$ has **no real solutions**. \n\nThus, we only need to solve:\n\n$$\\sin x = \\frac{1}{2}$$\n\nfor $-180^\\circ \\le x \\le 180^\\circ$."},{"type":"text","order":4,"title":"Find the final solutions","content":"We solve $\\sin x = \\frac{1}{2}$ within the interval $[-180^\\circ, 180^\\circ]$.\n\n* **Principal Value:** \n $$x = \\arcsin\\left(\\frac{1}{2}\\right) = 30^\\circ$$\n\n* **Second Value:** \n Using the symmetry of the sine graph ($180^\\circ - x$):\n $$x = 180^\\circ - 30^\\circ = 150^\\circ$$\n\nBoth $30^\\circ$ and $150^\\circ$ lie within the specified range $[-180^\\circ, 180^\\circ]$.\n\nTherefore, the solutions are:\n\n$$x = 30^\\circ, 150^\\circ$$"}]},{"id":"9e48ba86-fda2-4e28-9006-ef63a34c5114:graphical","label":"Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the function","content":"To solve the equation $\\cos 2x - 7 \\sin x + 3 = 0$ graphically, we can define the left-hand side of the equation as a function of $x$:\n\n$$y = \\cos 2x - 7 \\sin x + 3$$\n\nOur goal is to find the values of $x$ in the interval $[-180^\\circ, 180^\\circ]$ where this function is equal to zero (the $x$-intercepts of the graph)."},{"type":"text","order":1,"title":"Configure the calculator","content":"Before graphing, ensure your calculator is set to **Degree** mode since the interval is given in degrees ($-180^\\circ \\leq x \\leq 180^\\circ$). Adjust the window settings to display the correct interval for $x$ and a suitable range for $y$ to clearly see the intercepts:\n\n- **Xmin:** $-180$\n- **Xmax:** $180$\n- **Xscl:** $30$\n- **Ymin:** $-5$\n- **Ymax:** $15$","calculator":[{"gdc":{"expression":"f1(x) = cos(2x) - 7sin(x) + 3","instructions":"1. Change the angle mode to Degrees in Settings.\n2. Open the Graph app and enter f1(x) = cos(2x) - 7sin(x) + 3.\n3. Adjust the window settings to: X: [-180, 180] and Y: [-5, 15].\n4. Plot the graph."},"ti84":{"expression":"Y1 = cos(2X) - 7sin(X) + 3","instructions":"1. Press [MODE], highlight 'DEGREE', and press [ENTER].\n2. Press [Y=] and enter Y1 = cos(2X) - 7sin(X) + 3.\n3. Press [WINDOW] and set: Xmin = -180, Xmax = 180, Xscl = 30, Ymin = -5, Ymax = 15.\n4. Press [GRAPH] to plot."},"desmos":{"expression":"y = \\cos(2x) - 7\\sin(x) + 3","instructions":"1. Click on the wrench icon (Graph Settings) in the top-right corner.\n2. Select 'Degrees'.\n3. Set the X-axis limits to -180 to 180, and Y-axis limits to -5 to 15.\n4. Type the equation in the input bar."},"description":"Plot the function and set the viewing window in degree mode."}]},{"type":"text","order":2,"title":"Find the intercepts","content":"Look at the plotted graph. Notice that the curve intersects the $x$-axis at two points in the given domain $[-180, 180]$. Use the zero-finding feature of your graphing calculator to find these exact coordinates.","calculator":[{"gdc":{"instructions":"1. Go to Menu -> Analyze Graph -> Zero.\n2. Click to the left and right of each of the two x-intercepts to display their coordinates."},"ti84":{"instructions":"1. Press [2nd] [TRACE] (CALC) and select option 2: 'zero'.\n2. For the first zero, set the Left Bound to the left of the first intercept (e.g., x = 0), the Right Bound to its right (e.g., x = 50), and press [ENTER] on Guess. Repeat for the second intercept with bounds around x = 150.\n3. The calculator displays the solutions."},"desmos":{"instructions":"1. Click directly on the curve.\n2. Click on the grey dots that appear at the x-intercepts to reveal their coordinates."},"description":"Locate the x-intercepts of the curve."}]},{"type":"text","order":3,"title":"State final solutions","content":"The calculator yields the following $x$-intercepts within the interval $[-180^\\circ, 180^\\circ]$:\n\n$$x = 30^\\circ, \\quad x = 150^\\circ$$\n\nThese correspond to the exact solutions of our trigonometric equation."}]}],"generatedAt":"2026-06-09T04:09:16.952Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"1172b8ca-ac50-43dd-a696-2fbb942d3122","revisionId":"c9a6c3c5-a1e8-4688-8db6-09cb48694934","questionNumber":null,"questionType":"LA","specification":"","difficulty":null,"difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"6e8f40b1-8e1f-4b22-9a1c-9ab4a39dfd17","label":null,"content":"Solve $\\tan^2x=3$ for $0^\\circ\\leq x<360^\\circ$.","markscheme":"- $\\tan x=\\pm\\sqrt3$ M1\n- The reference angle is $60^\\circ$ M1\n- Positive tangent gives $60^\\circ,240^\\circ$ A1\n- Negative tangent gives $120^\\circ,300^\\circ$ M1\n- $x=60^\\circ,120^\\circ,240^\\circ,300^\\circ$ 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