Inverse Trigonometric Functions

Inverse Trigonometric Functions
Inverse circular function (Inverse Trigonometric Functions)

The functions sin^-1x, cos^-1x, tan^-1x, cot^-1x, cosec^-1x and sec^-1x are called inverse circular or inverse trigonometric functions. sin^-1x should not be confused with

(sin x)^-1 which is equal to .

Each of the inverse circular functions is multivalued. To make each inverse circular function single valued we define principal values as follows. If x is positive, the

principal values of all the inverse circular functions lie between 0 and . If x is negative, the principal values of sin^-1x, cosec^-1x and tan^-1x lie betweenand 0, and

those of cos^-1x, sec^-1x and cot^-1x lie between and . From now onwards we take only the principal values.

Hence sin = x = sin^-1x where (range) and x [-1, 1] (domain).

This ensures that the function = sin^-1x is one-one and onto.

Similarly cosec = x = cosec^-1x where and x (-, -1] [1, ).

cos = x = cos^-1x where [0, ] and x [-1, 1].

sec = x = sec^-1x where and x (-, -1] [1, ).

tan = x = tan^-1x where and x (-, ).

cot = x = cot^-1x where (0, ) and x (-, ).


The domains and ranges of inverse trigonometric (or inverse circular) functions are:

Function	Domain	Range
sin-1 x	[- 1, 1]
cos-1 x	[- 1, 1]	[0, ]
tan-1 x	R
cot-1 x	R	(0, )
sec-1 x	(- , - 1] [1, )
cosec-1 x	(- , - 1] [1, )

The graphs of = sin^-1x, = cos^-1x and = tan^-1x are:



Note that
(i) sin^-1(sin) = if and only if - and sin(sin^-1x) = x where -1 x 1.

(ii) cosec^-1(cosec) = if and only if -< 0 or 0 <

and cosec(cosec^-1x) = x where - < x -1 or 1 x < .

(iii) tan^-1(tan) = if and only if -<< and tan(tan^-1x) = x where - < x < .

(iv) cos^-1(cos) = if and only if 0 and cos(cos^-1x) = x where -1 x 1.

(v) sec^-1(sec) = if and only if 0 < or <

and sec(sec^-1x) = x where - < x -1 or 1 x < .

(vi) cot^-1(cot) = if and only if 0 << and cot(cot^-1x) = x where - < x < .


Some Important Results

sin^-1(-x) = - sin^-1(x)


cosec^-1(-x) = -cosec^-1(x)


tan^-1(-x) = -tan^-1(x)


cos^-1(-x) = - cos^-1(x)


sec^-1(-x) = - sec^-1(x)


cot^-1(-x) = - cot^-1(x)











Within the domain of their definition
sin^-1x = = cosec^-1= cos^-1


= , x > 0




= cosec^-1 = cot^-1 x > 0





cot^-1 x = cos^-1 = sec^-1

, x 0, y 0


2cos^-1x = cos^-1 (2x² - 1), x > 0


cos^-1x - cos^-1y =



sin^-1x + sin^-1y =


2sin^-1x =


sin^-1x - sin^-1y = sin^-1 , x 0, y 0


tan^-1x + tan^-1y =


x > 0, y > 0

2tan^-1x = , x 0