a1ef2ebe5e","label":"a","content":"When $\\theta$ is small, show that $\\frac{3\\cos \\theta}{\\sin \\theta}$ can be approximated by $\\frac{6 - 3\\theta^2}{2\\theta}$","markscheme":"- Substitutes $\\cos \\theta \\approx 1 - \\frac{\\theta^2}{2}$ and $\\sin \\theta \\approx \\theta$ M1\n- Obtains $\\frac{3\\left(1 - \\frac{\\theta^2}{2}\\right)}{\\theta}$ M1\n- Multiplies numerator and denominator by 2 to reach $\\frac{6 - 3\\theta^2}{2\\theta}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Substitute the small angle approximations for cos and sin.","code":"M"},{"id":"p2","marks":1,"description":"Form the initial fractional expression using the approximations.","code":"M"},{"id":"p3","marks":1,"description":"Correct algebraic simplification to the required form.","code":"A"}],"finalAnswers":["\\frac{6 - 3\\theta^2}{2\\theta}"],"generatedAt":"2026-05-06T14:04:58.741Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"direct_substitution","label":"Direct Substitution","steps":[{"type":"text","order":0,"title":"Recall small-angle approximations","content":"For a small angle $\\theta$ measured in radians, we use the standard small-angle approximations from our formula booklet:\n\n$$\\cos \\theta \\approx 1 - \\frac{\\theta^2}{2}$$\n\n$$\\sin \\theta \\approx \\theta$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$3\\cos\\theta \\approx 3\\htmlData{anchor=cos-approx}{\\left(1 - \\frac{\\theta^2}{2}\\right)}$$"},{"latex":"$$$\\sin\\theta \\approx \\htmlData{anchor=sin-approx}{\\theta}$$"},{"latex":"$$$\\frac{3\\cos \\theta}{\\sin \\theta} \\approx \\frac{3\\htmlData{anchor=numerator-sub}{\\left(1 - \\frac{\\theta^2}{2}\\right)}}{\\htmlData{anchor=denominator-sub}{\\theta}}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"cos-approx"},{"color":"amber","style":"box","anchor":"sin-approx"}],"connections":[{"to":"numerator-sub","from":"cos-approx","color":"sky","label":"substitute cos","style":"solid"},{"to":"denominator-sub","from":"sin-approx","color":"amber","label":"substitute sin","style":"solid"}]},"order":1,"title":"Substitute approximations","description":"We directly substitute these approximations into our expression $\\frac{3\\cos \\theta}{\\sin \\theta}$."},{"type":"text","order":2,"title":"Simplify the fraction","content":"To get rid of the fraction in the numerator, we multiply both the numerator and the denominator by $2$:\n\n$$\\frac{3\\left(1 - \\frac{\\theta^2}{2}\\right)}{\\theta} = \\frac{2 \\times 3\\left(1 - \\frac{\\theta^2}{2}\\right)}{2 \\times \\theta}$$\n\nNow, expand the numerator:\n\n$$= \\frac{6\\left(1 - \\frac{\\theta^2}{2}\\right)}{2\\theta}$$\n\n$$= \\frac{6 - 3\\theta^2}{2\\theta}$$\n\nThis completes the proof."}]}],"generatedAt":"2026-06-09T03:28:43.943Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"9c4c3439-9a79-435f-805e-e45a2ba621b9","label":"b","content":"Hence, approximate the value of $\\frac{3\\cos 0.2}{\\sin 0.2}$","markscheme":"- Substitutes $\\theta = 0.2$ into the approximation: $\\frac{6 - 3(0.2)^2}{2(0.2)}$ M1\n- Evaluates to $14.7$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Substitution of theta = 0.2 into the approximation formula.","code":"M"},{"id":"p2","marks":1,"description":"Correct calculation of the numerical value.","code":"A"}],"finalAnswers":["14.7"],"generatedAt":"2026-05-06T14:04:58.741Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"direct_substitution","label":"Substitution into Derived Approximation","steps":[{"type":"text","order":0,"title":"Identify the approximation formula","content":"In the previous part of the question, we derived a small-angle approximation for the expression $\\frac{3\\cos \\theta}{\\sin \\theta}$:\n\n$$\\frac{3\\cos \\theta}{\\sin \\theta} \\approx \\frac{6 - 3\\theta^2}{2\\theta}$$\n\nOur task now is to approximate the value of this expression when

Radians

Question 3

Radians Questions

Radians

Question 3

Radians Questions