A-level · Trigonometry

Small angle approximationsExam-style questions

The radian approximations sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − ½θ², why they hold, and using them to approximate expressions and evaluate limiting ratios for small θ.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4
  1. 1
    When θ\theta is small, show that 2sin⁡θ+3cos⁡θ≈3+2θ−32θ22\sin\theta + 3\cos\theta \approx 3 + 2\theta - \tfrac32\theta^2.
    Level 3 · 2 marksTry it
  2. 2
    When θ\theta is small, show that sin⁡3θtan⁡2θ≈32\dfrac{\sin 3\theta}{\tan 2\theta} \approx \dfrac{3}{2}.
    Level 3 · 2 marksTry it
  3. 3
    When θ\theta is small, show that cos⁡θ−1θ2≈−12\dfrac{\cos\theta - 1}{\theta^2} \approx -\dfrac{1}{2}.
    Level 3 · 2 marksTry it
  4. 4
    Use a small-angle approximation to estimate cos⁡0.1\cos 0.1 (where 0.10.1 is in radians), giving your answer to 4 decimal places. State the calculator value and comment.
    Level 3 · 3 marksTry it
  5. 5
    When θ\theta is small, show that (1+tan⁡θ)(1−sin⁡θ)≈1−θ2(1 + \tan\theta)(1 - \sin\theta) \approx 1 - \theta^2.
    Level 3 · 2 marksTry it

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