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 sinθ\sin \theta?

sinθθ\sin \theta \approx \theta

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

cosθ1θ22\cos \theta \approx 1 - \frac{\theta^2}{2}

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

tanθθ\tan \theta \approx \theta

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

Radians

Why does tanθθ\tan \theta \approx \theta for small angles?

sinθθ\sin \theta \approx \theta and cosθ1\cos \theta \approx 1

In the Taylor series for sinθ=θθ33!+...\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 sinθθ\frac{\sin \theta}{\theta} approximate to?

11

What is the first correction term in the cosine approximation cosθ1?\cos \theta \approx 1 - ?

θ22\frac{\theta^2}{2}

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