estionNumber":"3","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":7,"sourceTitle":"5.5 Small Angle Approximations","calculatorUse":null,"parts":[{"id":"05bdb45e-2f53-4d81-b0f5-80519b595996","label":"a","content":"Given that $\\theta$ is small, use the small angle approximation of $\\cos \\theta$ to show that\n\n$$4 \\cos(\\theta) + \\cos^2(2\\theta) \\approx 5 - 6\\theta^2 + 4\\theta^4$$","markscheme":"- Substitutes $4(1 - \\frac{\\theta^2}{2}) + (1 - \\frac{(2\\theta)^2}{2})^2$ M1\n- Simplifies to $4 - 2\\theta^2 + (1 - 2\\theta^2)^2$ M1\n- Expands $(1 - 2\\theta^2)^2$ to $1 - 4\\theta^2 + 4\\theta^4$ and collects terms to reach $5 - 6\\theta^2 + 4\\theta^4$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly applies the cosine small angle approximation to both terms.","code":"M"},{"id":"p2","marks":1,"description":"Correctly simplifies the squared term before expansion.","code":"M"},{"id":"p3","marks":1,"description":"Expands and collects terms correctly to reach the given quartic expression.","code":"A"}],"finalAnswers":["5 - 6\\theta^2 + 4\\theta^4"],"generatedAt":"2026-05-06T03:40:34.600Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"direct_substitution","label":"Direct Small-Angle Substitution","steps":[{"type":"text","order":0,"title":"Recall the approximation formula","content":"When $\\theta$ is measured in radians and is small, we can approximate trigonometric functions using algebraic terms. From your formula booklet, the small-angle approximation for cosine is:\n\n$$\\cos(x) \\approx 1 - \\frac{x^2}{2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4 \\cos(\\htmlData{anchor=t1}{\\theta}) + \\cos^2(\\htmlData{anchor=t2}{2\\theta})$"},{"latex":"$$4\\left(1 - \\frac{\\htmlData{anchor=t1-sub}{\\theta}^2}{2}\\right) + \\left(1 - \\frac{(\\htmlData{anchor=t2-sub}{2\\theta})^2}{2}\\right)^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t1"},{"color":"amber","style":"text-color","anchor":"t2"}],"connections":[{"to":"t1-sub","from":"t1","color":"sky","label":null,"style":"solid"},{"to":"t2-sub","from":"t2","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Substitute into the expression","description":"We apply the approximation directly to both $\\cos(\\theta)$ and $\\cos(2\\theta)$ by replacing $x$ with $\\theta$ and $2\\theta$ respectively. This earns the first mark M1."},{"type":"text","order":2,"title":"Simplify the inner terms","content":"Now we simplify each part of the expression before expanding:\n\n- **First term:**\n $$4\\left(1 - \\frac{\\theta^2}{2}\\right) = 4 - 2\\theta^2$$\n\n- **Second term:**\n $$1 - \\frac{(2\\theta)^2}{2} = 1 - \\frac{4\\theta^2}{2} = 1 - 2\\theta^2$$\n\nSubstituting these simplified terms back into our expression, we get:\n\n$$(4 - 2\\theta^2) + (1 - 2\\theta^2)^2$$\n\nThis simplified expression earns the second mark M1."},{"type":"text","order":3,"title":"Expand and collect terms","content":"Next, we expand the squared term $(1 - 2\\theta^2)^2$ using the identity $(a - b)^2 = a^2 - 2ab + b^2$:\n\n$$(1 - 2\\theta^2)^2 = 1 - 4\\theta^2 + 4\\theta^4$$\n\nNow, plug this back in and collect the like terms:\n\n$$\\begin{aligned} 4\\cos(\\theta) + \\cos^2(2\\theta) &\\approx (4 - 2\\theta^2) + (1 - 4\\theta^2 + 4\\theta^4) \\\\ &= (4 + 1) + (-2\\theta^2 - 4\\theta^2) + 4\\theta^4 \\\\ &= 5 - 6\\theta^2 + 4\\theta^4 \\end{aligned}$$\n\nThis completes the proof and earns the final mark A1."}]},{"id":"double_angle_identity","label":"Double Angle Identity First","steps":[{"type":"text","order":0,"title":"Understand the Goal","content":"We want to show that for small values of $\\theta$:\n\n$$4 \\cos(\\theta) + \\cos^2(2\\theta) \\approx 5 - 6\\theta^2 + 4\\theta^4$$\n\nFollowing the **Double Angle Identity First** approach, we will first rewrite the double-angle term $\\cos(2\\theta)$ using a double-angle identity, and then ap

Radians

Question 10

Radians Questions

Radians

Question 10

Radians Questions