Functions

Domain

Range
(Principal Value Ranges)


[1, 1]



[1, 1]



R (1, 1)



R (1, 1)



R



R



The value of inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions.

y = sin^{–1} x ⇒ x = sin y

x = sin y ⇒ y = sin^{–1} x

sin (sin^{–1} x) = x

sin^{–1}(sin x) = x


cos^{–1} (–x) = π – cos^{–1} x


cot^{–1} (–x) = π – cot^{–1} x


sec^{–1} (–x) = π – sec^{–1} x

sin^{–1}(–x) = – sin^{–1} x

tan^{–1} (–x) = – tan^{–1} x

tan^{–1} x + cot^{–1} x =

cosec^{–1} (–x) = – cosec^{–1} x

sin^{–1} x + cos^{–1} x =

cosec^{–1} x + sec^{–1} x =




Examples
Question
Find the principal value of
Solution
= y
We know that the range of the principal value branch of is, and
Hence the value of is
Question
Show that
Solution
Let Then
=
=