Mathematics –Class XII
Chapter-2: Inverse trigonometric
functions
Sub Topic: Properties of Inverse
trigonometric functions
Outline
Inverse Trigonometric function – Properties
Always remember to keep the constraint of domain and range , while solving inverse trigonometric functions.
sin( sin-1 x) = x if x is in [-1,1]
cos(cos-1 x) = x if x is in [-1,1]
tan( tan-1 x) = x x if x is in ( -∞, ∞)
sin-1 ( sin x) = x if x is in [-π/2, π/2]
cos-1 ( cos x) = x if x is in [0, π]
sec-1 ( sec x) = x if x is in [0, π] excl. x = π/2
Inverse Trigonometric function – Properties
cos-1 (-x) = π- cos-1x if x is in [-1,1]
sin-1 (-x) = -sin-1 (x) if x is in [-1,1]
tan-1(-x) = - tan-1x if x is in ( -∞, ∞)
cot-1(-x) = π - cot-1x if x is in (-∞,∞)
cosec-1(-x) = - cosec-1x if x is in (-∞,-1] U [1,∞)
sec-1(-x) = π - sec-1x if x is in (-∞,-1] U [1,∞)
Inverse Trigonometric function – Properties
Always remember to keep the constraint of domain and range , while solving inverse trigonometric functions.
sin-1 (-x) = -sin-1 (x) if x is in [-1,1]
Let y = sin-1(-x) ; constraint : y is in [-π/2, π/2]
Class Exercise :
Find the principal value of
Solution :
Class Exercise - 2
Find the principal value of sin –1 ( sin 5 )
Let y = sin-1(sin 5).Hence y is in [-π/2,π/2]
5
Wrong
sin 5 = sin ( 5 - 2 π)
sin-1(sin 5) = sin-1 ( sin ( 5 - 2 π))
= 5 - 2 π
Solution :
Other important properties
If x > 0 , y > 0 and xy < 1
If x > 0 , y > 0 and xy > 1
If x<0,y<0 and xy < 1
sin-1 x+ cos-1 x = π/2 ;
if x is in [-1,1]
Class Exercise - 5
Find the value of
Solution :
Class Exercise - 5
Find the value of
Solution :
Class Exercise - 8
If sin-1 x + sin-1 (1- x) = cos-1x,
the value of x could be
(a) 1, 0 (b) 1,1/2 (c) 0,1/2 (d) 1, -1/2
Solution :
Other important properties
Other important properties
Other important properties
ASSIGNMENTS
Q No.1-Prove that
Q.NO.2-solve for x:
Q No.3-Solve for x : 2tan-1(sinx)=tan-1(2secx),x≠π/2
Q No.4-Prove that
Q No.5-Prove that
Q No.6-Prove that
.
Q No.7-Show that
ASSIGNMENTS