Small-Angle Approximations

sin x is close to x

With x in radians, $\frac{\sin x}{x}\to 1$(sin x)/x → 1 as $x\to 0$x → 0. So for small x, sin x and x are nearly the same number, and $\sin x\approx x$sin x ≈ x. This is the first small-angle approximation.

On a graph, the sine curve leaves the origin running along the line y = x. They stay close for a while and then part: by x = 1 the curve is at sin 1 = 0.841, well below the line.

The gold curve y = sin x and the dashed line y = x, from x = −1.5 to 1.5. Near the origin they almost coincide; toward the ends the curve bends away below the line on the right and above it on the left.

cos x is close to $1-\frac{x^2}{2}$1 − x²/2

Near 0, cos x is close to 1, but $\cos x\approx 1$cos x ≈ 1 throws away how it changes. The double-angle formula $\cos 2A=1-2{\sin }^{2}A$cos 2A = 1 − 2 sin² A, with $A=\frac{x}{2}$A = x/2, gives $\cos x=1-2{\sin }^2\left(\frac{x}{2}\right)$cos x = 1 − 2 sin²(x/2). For small x, $\sin \left(\frac{x}{2}\right)\approx \frac{x}{2}$sin(x/2) ≈ x/2, so $\cos x\approx 1-2{\left(\frac{x}{2}\right)}^2=1-\frac{x^2}{2}$cos x ≈ 1 − 2(x/2)² = 1 − x²/2.

The parabola $y=1-\frac{x^2}{2}$y = 1 − x²/2 has its maximum, 1, at x = 0, where it is flat, and it falls away on both sides. Cosine has the same shape there: flat at its maximum, then falling.

The gold curve y = cos x and the dashed parabola $y=1-\frac{x^2}{2}$y = 1 − x²/2. Both are flat at height 1 at x = 0. At x = 1 they are 0.540 and 0.5, and by x = 1.5 they are 0.071 and −0.125.

tan x is close to x

$\tan x=\frac{\sin x}{\cos x}$tan x = (sin x)/(cos x). For small x the numerator is close to x and the denominator is close to 1, so $\tan x\approx \frac{x}{1}=x$tan x ≈ x/1 = x.

tan x runs above the line y = x on the right, and sin x runs below it: at x = 0.5, sin x = 0.479 and tan x = 0.546. Both are close to x, from opposite sides.

The gold curve y = tan x and the dashed line y = x, from x = −1.2 to 1.2. They leave the origin together; on the right the curve climbs above the line, on the left it falls below it.

How close, and for which x

At x = 0.1, sin 0.1 = 0.0998334, so $\sin x\approx x$sin x ≈ x is out by 0.000167, about 0.17%, and the two agree to three decimal places. tan 0.1 = 0.1003347, out by 0.000335, about 0.33%. cos 0.1 = 0.9950042 against 1 − 0.005 = 0.995, out by 0.0000042.

At x = 0.5 the errors grow to 0.021 for sine (4.3%), 0.046 for tangent (8.5%) and 0.0026 for cosine (0.29%). At x = 1 they are 19%, 36% and 7.5%.

Each is within 1% for |x| below about 0.244 radians for sine, 0.173 for tangent and 0.662 for cosine, which are about 14°, 9.9° and 38°. So the approximations are for angles of a few degrees, measured in radians. Cosine’s stays within 1% the longest.

Radians only

Every one of these rests on $\frac{\sin x}{x}\to 1$(sin x)/x → 1, and that limit holds only with x in radians. In degrees, $\sin x\approx x$sin x ≈ x fails badly: 0.1° is 0.001745 radians, so sin 0.1° = 0.001745, not 0.1. An angle given in degrees has to be converted, by multiplying by $\frac{\pi }{180}$π/180, before any of the approximations is used.

Using them together

For small x, $\frac{1-\cos 2x}{x\tan x}$(1 − cos 2x)/(x tan x) can be estimated by replacing each part. With 2x in place of x, $\cos 2x\approx 1-\frac{{\left(2x\right)}^2}{2}=1-2x^2$cos 2x ≈ 1 − (2x)²/2 = 1 − 2x², so the numerator is about $2x^2$2x². $\tan x\approx x$tan x ≈ x, so the denominator is about $x^2$x². The quotient is about $\frac{2x^2}{x^2}=2$2x²/x² = 2. Check: at x = 0.1 it is 1.98669, and at x = 0.05 it is 1.99667.

Here $\cos 2x\approx 1$cos 2x ≈ 1 would not do: the numerator would come out as 0. When an answer is a difference such as 1 − cos x, the approximation must keep the first term that does not cancel, which is the $x^2$x² term.

The usual mistakes

Using the approximations in degrees. They hold only for x in radians.

Writing $\cos x\approx 1+\frac{x^2}{2}$cos x ≈ 1 + x²/2. The plus sign bends the parabola upward, but near 0 cosine falls away from 1; it is never above 1.

Writing $\cos x\approx 1-x$cos x ≈ 1 − x. A straight drop falls at gradient −1 from x = 0, but cosine leaves x = 0 flat.

Writing $\tan x\approx 1$tan x ≈ 1. tan 0 = 0; it is the denominator, cos x, that is close to 1.

Using them for angles that are not small. At x = 1 radian, about 57°, $\sin x\approx x$sin x ≈ x is out by 19%.

A surveyor’s small angles

In the application below, an angle of 0.02 radians gives a mast’s height through $\tan \theta \approx \theta$tan θ ≈ θ, and an angle of 0.00125 radians gives the drop of a level sight line through $\cos \theta \approx 1-\frac{{\theta }^2}{2}$cos θ ≈ 1 − θ²/2, where $\cos \theta \approx 1$cos θ ≈ 1 would give no drop at all.