Derivative of Inverse Of trigonometric function

Derivative of Inverse Of trigonometric function
If u(x) is differential function of x y sin 1(x) x sin( y)
Iff
= ? ? =
2
1
1 ( )
sin ( )
u
u
dx
y u dy
?
?
= ? ? =
2
1
1 ( )
cos ( )
u
u
dx
y u dy
?
? ?
= ? ? =
2
1
1 ( )
tan ( )
u
u
dx
y u dy
+
?
= ? ? =
2
1
1 ( )
cot ( )
u
u
dx
y u dy
+
? ?
= ? ? =
( ) 1
sec ( )
2
1
?
?
= ? ? =
u u
u
dx
y u dy
( ) 1
csc ( )
2
1
?
? ?
= ? ? =
u u
u
dx
y u dy
Some Important Properties the inverse of trigonometric function
1 sin?1(?x) = ?sin?1(x) cos?1(?x) =? ? cos?1(x) tan?1(?x) = ?tan?1(x)
2 cot?1(?x) = ?cot?1(x) sec?1(?x) =? ? sec?1(x) csc?1(?x) = ?csc?1(x)
3
cos ( )
2
sin?1(?x) = ? ?1 x
?
cot ( )
2
tan?1(?x) = ? ?1 x
?
csc ( )
2
sec?1(?x) = ? ?1 x
?
4
sin 1( ) csc 1(1)
x
? x = ? cos 1( ) sec 1(1)
x
? x = ? tan 1( ) cot 1(1)
x
? x = ?
5 sin?1[sin(x)] = x sin[sin?1(x)] = x
Some Important Properties the inverse of Hyperbolic function
1 cosh2 (x) ? sinh2 (x) = 1 tanh2 (x) + sech2 (x) = 1 coth2 (x) ? csch2 (x) = 1
2 cosh(?x) = cosh(x) sinh(?x) = ?sinh(x) tanh(?x) = ? tanh(x)
3 sinh(x ± y) = sinh(x) cosh( y) ± cosh(x)sinh( y) cosh(x) ± sinh(x) = emx
4 cosh(x ± y) = cosh(x)cosh(y) m sinh(x)sinh( y) sinh(2x) = 2sinh(x)cosh(x)
Show that cosh2 (x) ? sinh2 (x) = 1
Solution:-
1
4
4
4
2
4
2
2 2
cosh ( ) sinh ( )
2 2 2 2 2 2
2 2 = =
? +
?
+ +
= ?
?
?
?
? ?
?
? ?
? ?
?
?
?
? ?
?
? +
? =
ex e?x ex e?x e x e? x e x e? x x x
Show that 1- sin?1(?x) = ?sin?1(x) 2- cos ( )
2
sin?1(?x) = ? ?1 x
?
Solution:-
1- Let y = sin?1(?x)
? ? x = sin(y) ? x = ?sin(y) ? x = sin(?y) ? ? y = sin?1(x) ?y = ?sin?1(x)
sin( )
sin 1( ) 1
x
? x ?
2- Let y = sin?1(?x)
cos ( )
2
cos ( )
2
)
2
? ? x = sin( y) ? x = cos( ? y ? ? y = ?1 x ? y = ? ?1 x
? ? ?
 cos ( )
2
y = ? ?1 x
?
If ?
?
?
?
? ?
?
?
= ?
2
3
x sin 1 Find cos(x) , tan(x)
Solution:- Let ?
??
?
? ??
?
= ?
2
x sin 1 3  sin ( x ) = ? ?
?
?
? ?
?
?
2
3
2
1
4
1
4
1 3
2
cos ( ) sin ( ) 1 cos( ) 1 sin ( ) 1 3
2
2 2 2 = ? = = ?
??
?
? ??
?
x + x = ? x = ? x = ?
Find the value of x If
4
tan 1( ) cot 1( ) ?
? x ? ? x =
Solution:-
??
?
??
= ? = + = ? = ? = ? ??
?
??
? ? ? ? ? ? ? ?
8
tan 3
8
tan ( ) 3
4
3
4 2
2 tan ( )
4
tan ( )
2
tan 1 1 1 1 ? ? ? ? ? ? ? x x x x