Inverse Trig Functions

Definition of inverse trig functions

To find the inverse trig functions, we have to know that trig functions are periodic functions. In the case of , there are many values of x: such that so that for the trigonometric functions to be defined, x must be restricted within a certain interval. From this, we can derive the inverse trig functions.

Trig functions are defined as:

When it's at x = , sin x = - 1 .

When it's at x = , sin x = 1 .

Simply speaking, it is the shortest ranges of ? that allow for the maximum ranges of values for the respective trigonometry function. This is the same for the other functions.

The inverse trig functions are defined as such:

If x is out of the range of -1 and 1, sine x and cosine x would be undefined as the trigonometry graphs would not cut the horizontal line of

inverse trig function for cosine


Scholar's note : Inverse trig functions are different from reciprocal functions, i.e.

wrong application of inverse trig functions

Principal Values of Inverse Trig Functions

The principal value of

a) is the value of ? in the interval [ ] such that

b) is the value of ? in the interval such that

c) is the value of ? in the interval such that

When the horizontal line y = k cuts the graph of a trigonometry function, we are finding the principle value of ? within their respective intervals.

For example, the principal value of is and of is


Here is an example of how inverse trig functions are used.

Evaluate without the use of tables or calculators problem on inverse trig functions.

picture showing the ratio of the inverse trig ratios


Let A= and B=

and the principle values of

and the principle values of

Then we make use of addition formula to expand cos(A+B)

We can find the values for the trigonometry ratios for them by drawing out the triangles.


We hope you understood how to apply inverse trig functions into trig problems.

Return to Trigonometry Help or Basic Trigonometry .