Download 5.6: Inverse Trigonometric Functions – Differentiation

5.6.1
5.6: Inverse Trigonometric Functions – Differentiation
Because none of the trigonometric functions are one-to-one, none of them have an inverse
function. To overcome this problem, the domain of each function is restricted so as to produce a
one-to-one function.
Inverse sine function:
y  sin 1 x
if and only if
  
sin y  x and y    , 
 2 2
Properties of the inverse sine function:
sin 1  sin x   x for 

2
x

2
sin  sin 1 x   x for 1  x  1
Example 1:
 3
1
1 
Evaluate sin 1 
 and sin    .
 2
 2 
5.6.2
Example 2:
3

Evaluate cot  sin 1 
7

Example 3:
Evaluate sin  sin 1  0.54   .
Example 4:
Evaluate sin  sin 1 2  .
Example 5:
 
Evaluate sin 1  sin 
4

Example 6:
 5 
Evaluate sin 1  sin

6 

5.6.3
Inverse cosine function:
y  cos 1 x
if and only if
Properties of the inverse cosine function:
cos 1  cos x   x for 0  x  
cos  cos 1 x   x for 1  x  1
Example 7:
 1
Evaluate cos 1   
 2
Example 8:

 5
Evaluate cos 1  cos  
 6




cos y  x and y   0,  
5.6.4
Inverse tangent function:
y  tan 1 x
lim tan 1 x 
x 
if and only if

2
  
tan y  x and y   , 
2 2
and lim tan 1 x  
x 

2
Graph of y  tan 1 x :
Inverse secant, cosecant, and cotangent functions:
These are not used as often, and are not defined consistently. Our book defines them as follows:
y  cot 1 x
if and only if
cot y  x and y   0,  
y  csc 1 x
if and only if
    
csc y  x and y    , 0    0, 
 2   2
y  sec 1 x
if and only if
    
sec y  x and y  0,    ,  
 2 2 
An important identity:
cos 1 x  sin 1 x 

2
5.6.5
Differentiation of the inverse sine function:
Since y  sin x is continuous and differentiable, so is y  sin 1 x .
We want to find its derivative.
d
1
sin 1 x  

dx
1  x2
Example 9:
Differentiate f  x   sin 1  2 x  7  .
, 1  x  1
5.6.6
Differentiation of the inverse cosine function:
d
1
cos 1 x   

dx
1  x2
, 1  x  1
Differentiation of the inverse cosine function:
d
1
tan 1 x  

dx
1  x2
Derivatives of other inverse trigonometric functions:
d
1
sin 1 x  

dx
1  x2
d
csc1 x   

dx
x
1
x2 1
d
1
cos 1 x   

dx
1  x2
d
1
sec 1 x  

dx
x x2  1
d
1
tan 1 x  

dx
1  x2
d
1
cot 1 x   

dx
1  x2
5.6.7
Example 10: Find the derivative of f  x   e tan
1
2x
.
Example 11: Find the derivative of y  csc 1  tan x  .
Example 12: Find the derivative of f ( x)  x3 arccos 2 x .
Example 13: Find the equation of the line tangent to the graph of f ( x)  arctan x at the point
where x  1 .