generatedAt":"2026-06-09T05:35:34.356Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"8714d6b0-9b11-45c2-b4a2-7e1eab0cf698","revisionId":"28952a7c-8468-4d48-95f4-cfc0575503ea","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"c1e799ed-cf97-45ff-8f67-c23268123637","label":"a","content":"Solve, for $0 \\leq \\theta \\leq 360$, the equation,\n\n$$10 \\sin \\theta \\cos \\theta = -3.5$$\n\nGive your answers to 1 decimal place.","markscheme":"- Uses $2 \\sin \\theta \\cos \\theta = \\sin 2\\theta$ to rewrite as $5 \\sin 2\\theta = -3.5$ M1\n- $\\sin 2\\theta = -0.7$ M1\n- Finds values for $2\\theta$ in the range $[0, 720]$: $224.42, 315.58, 584.42, 675.58$ M1\n- Divides by 2 to find $\\theta$ M1\n- Final answers: $112.2, 157.8, 292.2, 337.8$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Uses $2 \\sin \\theta \\cos \\theta = \\sin 2\\theta$ to rewrite the equation.","code":"M"},{"id":"p2","marks":1,"description":"Obtains $\\sin 2\\theta = -0.7$.","code":"M"},{"id":"p3","marks":1,"description":"Finds at least two correct values for $2\\theta$ (e.g., $224.42, 315.58$).","code":"M"},{"id":"p4","marks":1,"description":"Divides the values of $2\\theta$ by 2 to find $\\theta$.","code":"M"},{"id":"p5","marks":1,"description":"Provides all four correct final answers to 1 decimal place: $112.2, 157.8, 292.2, 337.8$.","code":"A"}],"finalAnswers":["112.2, 157.8, 292.2, 337.8"],"generatedAt":"2026-05-06T14:03:56.858Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"double_angle","label":"Double Angle Identity","steps":[{"type":"text","order":0,"title":"Use double angle identity","content":"We begin with the given equation:\n\n$$10 \text{ sin } \theta \text{ cos } \theta = -3.5$$\n\nRecall the double angle identity for sine from the formula booklet:\n\n$$\\sin 2\\theta = 2 \\sin \\theta \\cos \\theta$$\n\nWe can rewrite the LHS of our equation by factoring out $5$ to make the identity visible:\n\n$$5(2 \\sin \\theta \\cos \\theta) = -3.5$$\n\nSubstituting $\\sin 2\\theta$ into the equation gives:\n\n$$5 \\sin 2\\theta = -3.5$$","highlights":[{"text":"10 \\sin \\theta \\cos \\theta = -3.5","location":"specification"}]},{"type":"text","order":1,"title":"Isolate the sine term","content":"To solve for $2\\theta$, divide both sides of the equation by $5$:\n\n$$\\sin 2\\theta = \\frac{-3.5}{5}$$\n\n$$\\sin 2\\theta = -0.7$$"},{"type":"text","order":2,"title":"Adjust the interval","content":"The original range for $\\theta$ is given as:\n\n$$0^\\circ \\le \\theta \\le 360^\\circ$$\n\nSince we are solving for $2\\theta$, we must multiply the interval limits by $2$ to find the range of valid angles for $2\\theta$:\n\n$$0^\\circ \\le 2\\theta \\le 720^\\circ$$","highlights":[{"text":"0 \\le \\theta \\le 360","location":"specification"}]},{"type":"text","order":3,"title":"Find principal value of 2θ","content":"Now, find the principal value (the calculator's inverse sine value) of $2\\theta$:\n\n$$2\\theta = \\sin^{-1}(-0.7) \\approx -44.43^\\circ$$\n\nSince $-44.43^\\circ$ is not in our interval $[0^\\circ, 720^\\circ]$, we will use it to find the positive angles within our target range.","calculator":[{"gdc":{"expression":"\\arcsin(-0.7)","instructions":"Ensure angle mode is set to degrees, then input asin(-0.7)."},"ti84":{"expression":"\\sin^{-1}(-0.7)","instructions":"1. Press [MODE] and ensure DEGREE is highlighted.\n2. Press [2nd] [SIN] to get sin⁻¹(\n3. Enter -0.7 and press [ENTER]."},"desmos":{"expression":"\\arcsin(-0.7)","instructions":"Set the calculator to Degrees by clicking the wrench icon, then type arcsin(-0.7)."},"description":"Calculate the inverse sine of -0.7 in degree mode."}]},{"type":"text","order":4,"title":"Find all values of 2θ","content":"Sine is negative in the 3rd and 4th quadrants. We can fi

Trigonometry and Modelling

Question 9

Trigonometry and Modelling Questions

Trigonometry and Modelling

Question 9

Trigonometry and Modelling Questions