Download Section 3.5 – Inverse Trig Functions

Chapter 3. Section 5
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Section 3.5 – Inverse Trig Functions
Recall the Trigonometric Functions:

The six trig functions are defined in terms of right triangles (see figure).

The two “most important” being sine and cosine.
opp.
hyp.
adj.
cos  
hyp.
sin  

All other trig functions can be defined in terms of sine and cosine, so remember the definitions
above, and the relationships between them and the others…
1
cos 
1
csc  
sin 
sec  
sin 
cos 
cos 
cot  
sin 
tan  

Evaluating trig functions boils down to some fundamentals.

Remember the basic shape of sin(x) and cos(x), along with when they are 0, and 1 .

From the figure you can see they are periodic, both with a period of 2.
The Inverse Trig Functions:

The inverse trig functions are defined to ‘undo’ the original trig functions. For example, we want
sin 1 (sin x)  x and sin(sin 1 x)  x .

Q: Take a look at the x’s above. Which represents an angle and which is a numeric value?
A:
C. Bellomo, revised 24-Aug-12
Chapter 3. Section 5
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
To find an inverse that is a function, we need to ensure the original function (for example y  sin x )
is one to one.

The trig functions are not inherently one to one. Therefore, we must restrict the domain of the
   
,
original function to be one to one. In the case of sine, we restrict it to be between 
 2 2 

Recall to find an inverse, we switch the x’s and y’s and solve for y.
The Inverse Trig Functions:


sin 1 x  y
iff
cos 1 x  y
iff
tan 1 x  y
iff


 y
2
2
cos y  x and 0  y  


tan y  x and
 y
2
2
sin y  x and
Q: What are the restrictions on the x values above?
A:
Derivatives of the Inverse Trig Functions:

To be able to determine the derivative of sine inverse, we first must recall some trig… If we want to
graph sin y = x we would have the following representation
This would mean that cos( y )  1  x 2
Using implicit differentiation, we find
d
(sin y  x)
dx
y cos y  1
1
1
y 

cos y
1  x2
d
1
(sin 1 x) 
dx
1  x2
C. Bellomo, revised 24-Aug-12
Chapter 3. Section 5
Page 3 of 4


Summary
d
1
(sin 1 x) 
dx
1  x2
d
1
(csc 1 x)  
dx
x 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
Q: What are the domain restrictions for the above functions?
A:
Some Example Problems:

 3
3
Example. Find the value of cos 1 
for x in [0, 2).
 and use it to solve cos x 
2
 2 

Example. Find the derivative of y  cos 1 (e x  2 x) .
C. Bellomo, revised 24-Aug-12
Chapter 3. Section 5
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
 x 1 
Example. Find the derivative of y  sin 1 

 x 1 
C. Bellomo, revised 24-Aug-12