Inverse Trigonometric Functions.

Let be the function defined on the interval by . The function is continuous and strictly increasing on , and . Therefore is invertible. Its inverse is called arcsinus and denoted .

For example: ; .

Remark 5.7.7   For any , we have , but generally . For example, .

Let be the function defined on the interval by . The function is continuous and strictly decreasing on , and . Therefore is invertible. Its inverse is called arccosinus and denoted .

For example: ; .

Remark 5.7.8   For any , we have , but generally . For example, .

Let be the function defined on the interval by . The function is continuous and strictly increasing on , and . Therefore is invertible. Its inverse is called arctangent and denoted .

For example: ; .

Remark 5.7.9   For any , we have , but generally . For example, .

The inverse trigonometric functions verify many interesting identities; let us see some of them as examples.

Example 5.7.10   We wish to simplify the expressions and , for any .
1. For any , the following identity holds:

Therefore

It follows that . As belongs to the interval , we know that . Finally we have:

 (5.1)

2. For any , the following identity holds:

Therefore

It follows that . As belongs to the interval , we know that . Finally we have:

 (5.2)

We will have an important usage of these identities in Chapter 6.
Noah Dana-Picard 2007-12-28