Trigonometric functions

Trigonometric functions
Definitions of trigonometric functions for a right triangle
A right triangle is a triangle with a right angle (90°)

For every angle in the triangle, there is the side of the triangle adjacent to it, the side opposite of it and the hypotenuse such that
For angle , the trigonometric functions are defined as follows:



Trigonometric functions of negative angles

Some useful relationships among trigonometric functions






Graphs of Trigonometric Functions


Derivatives of trigonometric functions
If is a function , the chain rule version of this differentiation rule is






Example 1: Find derivatives of the functions
















trigonometric functions Inverse
The inverse trigonometric functions are defined to be the inverses of particular parts of the trigonometric functions; parts that do have inverses.
The inverse sine function, denoted by (or ), is defined to be the inverse of the restricted sine function. A similar idea holds for all the other inverse
trigonometric functions. It is important here to note that in this case the “-1” is not an exponent and so,

In inverse trigonometric functions the “-1” looks like an exponent but it isn’t, it is simply a notation that we use to denote the fact that we’re dealing with an inverse trigonometric function. It is a notation that we use in this case to denote inverse trigonometric functions. If we had really wanted exponentiation to denote 1 over sine we would use the following.


Derivatives of inverse trigonometric functions Let be a function the derivatives of inverse trigonometric functions are:



Example 4: Find the derivative for








Find derivative in each of the following problems