Small angle approximations

The small angle approximations say that when an angle θ is small and measured in radians, sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2. They let you replace trig functions with simple polynomials, which makes awkward expressions easy to simplify and limits easy to estimate. They are part of A-level Maths in England.

The three small angle approximations

Two conditions matter. First, θ must be in radians: the approximations are simply wrong in degrees. Second, θ must be small, close to 0. The closer θ is to 0, the better they work.

At θ = 0.1, sin θ and tan θ are both within 0.4% of 0.1, and cos θ agrees with 1 − θ²/2 to five decimal places. By θ = 0.5 the sine and tangent approximations are clearly off.

Why the small angle approximations work

The unit circle picture

In a circle of radius 1, an angle of θ radians cuts off an arc of length exactly θ. The vertical line from the circle to the axis has length sin θ, and the line touching the circle at the right has length tan θ. Here θ is drawn at 25° so you can see all three; for smaller angles the three lengths squeeze together, so sin θ ≈ θ ≈ tan θ. That is also why the angle must be in radians: only then is the arc length equal to θ.

Where cos θ ≈ 1 − θ²/2 comes from

Use the double angle identity cos θ = 1 − 2sin²(θ/2), then the approximation for sine on the half angle:

The series view

The approximations are the first terms of the Maclaurin series sin θ = θ − θ³/6 + …, cos θ = 1 − θ²/2 + θ⁴/24 − … and tan θ = θ + θ³/3 + …. When θ is small, θ³ and higher powers are tiny, so dropping them changes very little. You can compare the series with the real functions in the Taylor Series calculator.

How small is small?

Here x is the angle in radians and y is how far θ is from sin θ, as a percentage. The error stays under 1% up to about 0.24 radians, roughly 14°, then grows quickly. For tan θ ≈ θ the 1% limit is about 0.17 radians (10°), and cos θ ≈ 1 − θ²/2 stays within 1% up to about 0.66 radians (38°), because its next term is θ⁴/24.

Using small angle approximations: worked examples

Example 1: an approximate value

Use a small angle approximation to estimate cos 0.2.

The true value is 0.98007, so the estimate is excellent.

Example 2: angles given in degrees

Estimate sin 3°. Convert to radians first: 3° = 3π/180 = π/60 ≈ 0.05236. So sin 3° ≈ 0.0524. The true value is 0.05234.

Converting between degrees and radians is the first step in many of these questions. Practise it here.

Example 3: simplifying an expression

Show that when θ is small, (1 − cos 2θ) / (θ sin θ) ≈ 2.

Replace each trig function, remembering that the angle in the cosine is 2θ:

Check with θ = 0.05: the exact value is 1.9992.

Example 4: an expression in terms of θ

For small θ, find an approximation for (4 sin θ − tan 2θ) / (1 − cos θ).

So the expression is approximately 4/θ. Notice tan 2θ ≈ 2θ, not 2tan θ.

Example 5: solving an equation

Find an approximate small positive solution of cos θ = 0.995.

The exact solution is 0.10004 radians.

Try one more before the common mistakes.

Small angle approximations in physics

Physicists use sin θ ≈ θ all the time. For a pendulum swinging through a small angle, it turns the equation of motion into simple harmonic motion, which gives the period T = 2π√(l/g). It is also used in optics and in the diffraction of light, where the angles are tiny.

Common mistakes

• Using degrees. sin 5 ≈ 5 is nonsense in degrees. Convert to radians first.
• Approximating cos θ as 1 when the question needs the θ² term. Use 1 − θ²/2 unless told otherwise.
• Forgetting to square the whole angle. cos 2θ ≈ 1 − (2θ)²/2 = 1 − 2θ², not 1 − 2θ²/2.
• Writing sin 3θ ≈ 3 sin θ. Apply the approximation to the whole angle: sin 3θ ≈ 3θ.
• Using the approximations for large angles. By 0.5 radians the sine approximation is already over 4% out.

Practise with a short timed paper

Trigonometry, including small angle approximations, comes up in A-level Maths pure papers. The daily A-level Maths: Algebra & Functions paper is a short timed paper of 8 questions, with a new one every day.

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