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Small Angle Approximation

Maths: Edexcel A Level Pure Maths, topic 5.2.6

Small Angle Approximation

We can approximate the values of trigonometric functions when the angle is very small in order to simplify calculations. We call these the small angle approximations.

Sine approximation

Sine approximation

  • When the angle θ\theta is close to zero and measured in radians, we can approximate the sine of this angle as:
  • sin⁡θ≈θ\sin\theta \approx \theta
Cosine approximation

Cosine approximation

  • When the angle θ\theta is close to zero and measured in radians, we can approximate the cosine of this angle as:
  • cos⁡θ≈1−θ22\cos\theta \approx 1-\frac{\theta^2}{2}
Tangent approximation

Tangent approximation

  • When the angle θ\theta is close to zero and measured in radians, we can approximate the tangent of this angle as:
  • tan⁡θ≈θ\tan\theta \approx \theta
Example

Example

  • What is the approximation of the following expression when θ\theta is small?
  • cos⁡4θ+sin⁡3θ\cos4\theta + \sin3\theta
  • What is the approximate value of the expression when θ\theta is small?
Approximations

Approximations

  • When θ\theta is small, cos⁡θ≈1−θ22\cos\theta \approx 1-\frac{\theta^2}{2} and sin⁡θ≈θ\sin\theta \approx \theta.
  • That means we can say that:
  • cos⁡4θ≈1−(4θ)22\cos4\theta\approx 1 - \frac{(4\theta)^2}{2}
  • sin⁡3θ≈3θ\sin3\theta \approx 3\theta
Substitute approximations in

Substitute approximations in

  • Substituting these approximations in, we have:
  • cos⁡4θ+sin⁡3θ≈1−(4θ)22+3θ\cos4\theta + \sin3\theta\approx 1-\frac{(4\theta)^2}{2}+3\theta
  • Simplifying this expression gives:
  • cos⁡4θ+sin⁡3θ≈1−8θ2+3θ\cos4\theta + \sin3\theta\approx 1 - 8\theta^2 + 3\theta
Approximate value

Approximate value

  • When θ\theta is small, any terms with factors of θ\theta will also be small.
  • That means we can disregard these terms when finding the value of the expression:
  • cos⁡4θ+sin⁡3θ≈1\cos4\theta + \sin3\theta\approx 1

Test yourself on Small Angle Approximation

  • If θ\theta is small, what is sin⁡θ\sin\theta approximately equal to?

  • If θ\theta is small, what is tan⁡θ\tan\theta approximately equal to?

  • If θ\theta is small, what is cos⁡θ\cos\theta approximately equal to?

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