Radian Measure (Edexcel A-Level Mathematics): Quizzes

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Small Angle Approximations
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For small angles θ\theta in radians, what is the approximation for sin⁡θ\sin \theta?

sin⁡θ≈θ\sin \theta \approx \theta

For small angles θ\theta in radians, what is the approximation for cos⁡θ\cos \theta?

cos⁡θ≈1−θ22\cos \theta \approx 1 - \frac{\theta^2}{2}

For small angles θ\theta in radians, what is the approximation for tan⁡θ\tan \theta?

tan⁡θ≈θ\tan \theta \approx \theta

In which units must θ\theta be measured for small angle approximations to work?

Radians

Why does tan⁡θ≈θ\tan \theta \approx \theta for small angles?

sin⁡θ≈θ\sin \theta \approx \theta and cos⁡θ≈1\cos \theta \approx 1

In the Taylor series for sin⁡θ=θ−θ33!+...\sin \theta = \theta - \frac{\theta^3}{3!} + ..., what is neglected?

Higher-order terms like θ36\frac{\theta^3}{6}

In physics, where are small angle approximations frequently used?

Pendulum motion

Using small angle approximations, what is sin⁡(0.1)\sin(0.1) radians approximately equal to?

0.10.1

For small θ\theta, what does sin⁡θθ\frac{\sin \theta}{\theta} approximate to?

11

What is the first correction term in the cosine approximation cos⁡θ≈1−?\cos \theta \approx 1 - ?

θ22\frac{\theta^2}{2}

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