Inverse Trigonometric Functions, Maths First, Institute of Fundamental Sciences, Massey University

Inverse Trigonometric Functions

Given an angle θ, we now know how to calculate the trig functions sin θ, cos θ and tan θ using a scientific calculator. Often, however, we need to be able to find the angle θ given the values of one of the trig functions sin θ, cos θ or tan θ.

To do this we need the inverse trigonometric functions:

If sin θ = x, then θ = sin^-1x.

If cos θ = x, then θ= cos^-1x.

And

if tan θ = x, then θ = tan^-1x.

Warning: The exponent -1 on the inverse trig functions does not mean the reciprocal. For example,

Instead it is the inverse function of the sine. That is sin^-1(sinθ) = θ and sin(sin^-1θ) = θ. To avoid confusion, we also write arcsin x, or simply asin x, for sin^-1x (read as arc sine of x).

Similar notation is used for other trig functions:

If sin θ = x, then θ = sin^-1x = arcsin x.

If cos θ = x, then θ = cos^-1x = arccos x.

And

if tan θ = x, then θ = tan^-1x = arctan x.

For example, in the following triangle sin θ = 4/5.