Small Angle Approximation

sin(x) ≈ x     when x is near 0.

The slope of sin(0) is 1. Because the sine function changes gradually, the slope near 0 is still close to 1. This property lets us use the line equation y = x instead of the complicated sine function. Provided that x is "small", the error will be negligible.

This graph shows how y = sin(x) compares with y = x, which has a slope of 1 everywhere.

Percent Error:

(A y-axis value of 1 is equivalent to 100%.)

As you can see, the percent error is barely significant when x is near 0. For example, at x = 0.25 the error is 1%. If the uncertainty in your data is larger than 1%, then you can use the small angle approximation up to that amount of error.

Why Does it Work?

The sine function can also be expressed using the Taylor series:

sin(x) = x - x³3! + x⁵5! - x⁷7! + x⁹9! - ... = ∞Σn = 0 x^{2n + 1}(2n + 1)!(-1)ⁿ

With each term added, the value becomes closer to the actual value of sin(x). Look at the first term. It's just x, which is exactly the same as the small angle approximation!