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:
|sin-1 x||[- 1, 1]|
|cos-1 x||[- 1, 1]||[0, ]|
|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:
(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 =
sin-1x - sin-1y = sin-1 , x 0, y 0
tan-1x + tan-1y =
x > 0, y > 0
2tan-1x = , x 0