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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.

25 min Video by Zeeshan Zamurred Radians

Edexcel A Level Maths: 5.5 Small Angle ApproximationWatch the full walkthrough 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

The approximations

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

sin θ \approx θ, tan θ \approx θ, cos θ \approx 1 - \frac{θ ^{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.

sin 2 θ \approx 2 θ

\frac{2 θ}{θ} = 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 $θ - \frac{θ ^{3}}{6} + \dots$ when $θ$ is in radians. In degrees they fail.

Formula recap

sin θ \approx θ

Small θ (radians).

tan θ \approx θ

Small θ (radians).

cos θ \approx 1 - \frac{θ ^{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.

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