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5.5PureStretch

Small Angle Approximations

When an angle θ (in radians) is small, sin θ, tan θ and cos θ can be replaced by simple polynomial approximations. These are used to estimate values and to study limits — and they only work in radians.

25 min Radians
Edexcel A Level Maths: 5.5 Small Angle ApproximationCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • State the small angle approximations
  • Apply them to estimate trig values
  • Use them in compound expressions
  • Understand why radians are required
1

The approximations

For small θ\theta (in radians): sin⁡θ≈θ\sin\theta \approx \theta, tan⁡θ≈θ\tan\theta \approx \theta, and cos⁡θ≈1−θ22\cos\theta \approx 1 - \tfrac{\theta^2}{2}.

sin⁡θ≈θ,tan⁡θ≈θ,cos⁡θ≈1−θ22\sin\theta \approx \theta, \quad \tan\theta \approx \theta, \quad \cos\theta \approx 1 - \tfrac{\theta^2}{2}
Valid for small θ in radians.
2

Using them

Substitute the approximations into an expression and simplify. They are most useful when an angle clearly tends to zero or is given as small.

Example 1Approximate sin⁡2θθ\dfrac{\sin 2\theta}{\theta} for small θ\theta.
1sin⁡2θ≈2θ\sin 2\theta \approx 2\theta.
22θθ=2\dfrac{2\theta}{\theta} = 2.
Answer
≈2\approx 2

Tip — Apply the approximation to the whole argument: sin(2θ) ≈ 2θ, not 2·sin θ.

3

Why radians?

These approximations come from the series expansions of sin⁡\sin, cos⁡\cos, tan⁡\tan, which only take the simple form θ−θ36+⋯\theta - \tfrac{\theta^3}{6} + \cdots when θ\theta is in radians. In degrees they fail.

Formula recap

sin⁡θ≈θ\sin\theta \approx \theta
Small θ (radians).
tan⁡θ≈θ\tan\theta \approx \theta
Small θ (radians).
cos⁡θ≈1−θ22\cos\theta \approx 1 - \tfrac{\theta^2}{2}
Small θ (radians).

Common mistakes to avoid

Using cos θ ≈ θ.
cos θ ≈ 1 − θ²/2; only sin and tan ≈ θ.
Applying the approximations in degrees.
They are only valid in radians.

Key takeaways

  • sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2 (radians).
  • Apply to the whole argument of the function.
  • Only valid in radians, for small θ.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. For small θ (radians), sin⁡θ≈\sin\theta \approx

    • θ\theta
    • 11
    • 1−θ221 - \tfrac{\theta^2}{2}
    • θ2\theta^2
    Show answer and explanation

    Answer: θ\theta

    Leading term.

  2. For small θ, cos⁡θ≈\cos\theta \approx

    • 1−θ221 - \tfrac{\theta^2}{2}
    • θ\theta
    • 11
    • 1−θ1 - \theta
    Show answer and explanation

    Answer: 1−θ221 - \tfrac{\theta^2}{2}

    Quadratic term.

  3. For small θ, tan⁡θ≈\tan\theta \approx

    • θ\theta
    • 11
    • θ2\theta^2
    • 1−θ221 - \tfrac{\theta^2}{2}
    Show answer and explanation

    Answer: θ\theta

    Same as sin to first order.

  4. sin⁡2θθ\dfrac{\sin 2\theta}{\theta} for small θ ≈

    • 22
    • 11
    • 12\tfrac12
    • 2θ2\theta
    Show answer and explanation

    Answer: 22

    sin 2θ ≈ 2θ.

  5. These approximations are only valid when θ is in:

    • radians
    • degrees
    • gradians
    • any unit
    Show answer and explanation

    Answer: radians

    From the radian series.

Next step

Keep practising

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