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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.

30 min Trigonometry

The big picture

The small quantity is the whole angle. For $\sin(3x)$ use $3x$ , and for $\cos(2x)$ use $1-(2x)^2/2$ . Keep the highest power requested and do not treat an approximation as an exact identity. A denominator may approach zero too. Work out the leading nonzero terms before dividing; replacing every cosine by 1 immediately can erase the information needed.

Edexcel A Level Maths: 5.5 Small Angle ApproximationCurated by Kepler, not produced by us. Watch it, then work through 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

The approximations

For small $\theta$ (in radians): $\sin\theta \approx \theta$ , $\tan\theta \approx \theta$ , and $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$ .

\sin\theta \approx \theta, \quad \tan\theta \approx \theta, \quad \cos\theta \approx 1 - \tfrac{\theta^2}{2}

Valid for small θ in radians.

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

\dfrac{\sin 2\theta}{\theta}

for small

\theta

\sin 2\theta \approx 2\theta

\dfrac{2\theta}{\theta} = 2

Answer

\approx 2

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

Why radians?

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

Approximate a composite expression

The small quantity is the whole angle. For $\sin(3x)$ use $3x$ , and for $\cos(2x)$ use $1-(2x)^2/2$ . Keep the highest power requested and do not treat an approximation as an exact identity.

A denominator may approach zero too. Work out the leading nonzero terms before dividing; replacing every cosine by 1 immediately can erase the information needed.

Example 1For small nonzero x in radians, approximate

\dfrac{1-\cos2x}{x\sin3x}

1-\cos2x\approx1-(1-2x^2)=2x^2

x\sin3x\approx x(3x)=3x^2

3Their ratio is approximately

2x^2/(3x^2)=2/3

Answer

2/3

, for x sufficiently close to zero.

Tip — A degree-mode calculator does not make a degree value into a valid input for these radian expansions.

Think like an examiner

Write the defining equations and show the substitution before evaluating.
A degree-mode calculator does not make a degree value into a valid input for these radian expansions.

Common misconceptions

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 results to remember

\sin\theta \approx \theta

Small θ (radians).

\tan\theta \approx \theta

Small θ (radians).

\cos\theta \approx 1 - \tfrac{\theta^2}{2}

Small θ (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.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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