Definition of sin
-1x
For x Î [-1, 1], if q is an angle whose sine is x, then we say that sine inverse x is q and write sin-1x = q.
For practical purposes, only principal values of sin-1x are considered.
sin-1x is a function of x with domain [-1,1] and range [-p/2, p/2].The graph of sin-1x is as shown in the figure below.
Definition of cos
-1 x
cosine inverse x is q and write cos-1x = q.
For practical purposes, only principal values of cos-1x are considered.
cos-1x is a function of x with domain [-1,1] and range [0, p].The graph of cos-1x is as shown in the figure below.
Definition of tan
-1x
In particular,
The graph of tan-1x is as shown in the figure below.
Definition of cot
-1x
For x Î R, if q is an angle whose cotangent is x, then we say that cotangent inverse of x is q and write cot-1x= q.
In particular,
For practical purposes, only principal values of cot-1x are considered.
Definition of sec
-1x
For x Î R - (-1, 1), if q is an angle whose secant is x, then we say that secant inverse of x is q and write sec-1x = q.
In particular,
For practical purposes, only principal values of sec-1x are considered.
sec-1x is a function of x with domain R-(-1,1) and range [0,p]-{p/2}.Definition of cosec
-1x
For x Î R - (-1,1), if q is an angle whose cosecant is x, then we say that cosecant inverse of x is q and write cosec-1 x = q.
In particular, if q Î [-p/2, p/2] - {0}, then q is called the principal value of cosec-1 x.
cosec-1x is a function of x with domain R-(-1,1) and range [-p/2, p/2]-{0}.
Note:
sin-1 x is only a symbol to denote sine inverse of x.
Table of domain and range of inverse trigonometric function
Prove the following identities:
Proof:
Let,
Example 1:
Solve the following equation:
Suggested answer:
Example 2:
Prove the following equation:
Suggested answer:
