Download 5.6: Inverse Trigonometric Functions: Differentiation

AP CALCULUS - AB
Section Number:
LECTURE NOTES
Topics: Inverse Trigonometric Functions: Derivatives
MR. RECORD
Day: 1 of 2
5.6
I will start this section out with a bold statement: None of the basic trigonometric functions has an inverse function.
This is true because none of the six are one-to-one. (That is to say, they are not strictly monotonic or that they fail
the horizontal line test.)
However, we can redefine the domains of each of these functions so that they all will have an inverse on their
restricted domains.
Definition of Inverse Trigonometric Functions
Function
Domain of inverse function
Range of inverse function
y  arcsin x iff sin y  x
1  x  1
y  arccos x iff cos y  x
y  arctan x iff tan y  x
1  x  1
 2  y  2
0  y 
  x  
  x  
 2  y  2
0  y 
y  arccot x iff cot y  x
y  arcsec x iff sec y  x
x 1
y  arccsc x iff csc y  x
x 1
 x  1 or
 x  1 or
x  -1
0  y   , y  2
x  -1
 2  y  2 , y  0
Note: The abbreviation “iff” refers to the biconditional “if and only if.”
Graphs of the Six Inverse Trigonometric Functions
y  arcsin x
y  arctan x
y  arccsc x
y
y

y


x




x

x















y  arccos x
y  arccot x
y  arcsec x
y

y
y





x
x
x



















Example 1: Evaluating Inverse Trigonometric Functions.
Evaluate each of the following
a. arcsin  12 
c. arctan 3
b. arccos(0)
d. arcsin(0.3)
Example 2: Solving an Equation with an Inverse Trigonometric Function.
Solve the following for x:
arctan  2 x  3  

4
Example 3: Finding the Exact Value of Expressions Involving Inverse Trigonometric Functions.
Find the exact value of each. Draw a picture to describe the situation.

 2 5 

 1 
a. tan cos 1    
b. sec sin1 
 
 3 


 5  
Example 4: Using Right Triangles.
Answer each of the following.
a. Given y  arcsin x , where 0  y 

2
, find cos y.
b. Given y  arcsec  x  , find tany.
THEOREM: Derivatives of the Six Inverse Trigonometric Functions
Let u be a function of x.
d
u
arcsinu 
dx
1  u2
d
u
arccos u 
dx
1  u2
d
u
arctanu 
dx
1  u2
d
u
arccot u 
dx
1  u2
d
u
arcsec u 
dx
u u2  1
d
u
arccsc u 
dx
u u2  1
Example 5: Differentiating Inverse Trigonometric Functions.
Find each of the following derivatives.
d
d
arcsin  2 x  
arctan  3x  
a.
b.
dx
dx 
c.
d 
arctan x 
dx 
Example 6: Derivatives That Can Be Simplified.
Differentiate and simplify: y  arcsin x  x 1  x 2 .
d.
d
arcsec  e2 x  

dx 
AP CALCULUS – AB
Section Number:
5.6
LECTURE NOTES
Topics: Inverse Trigonometric Functions: Derivatives
- Inverse Trig Function Analyisis
- Optimization Application
MR. RECORD
Day: 2 of 2
Example 7: Analyzing an Inverse Trigonometric Graph.
Analyze the graph of : y   arctan x  .
Be sure to find intervals of increasing/decreasing behavior and concavity as well as relative extrema,
points of inflection and any asymptotes.
2
Another look at Optimization.
Example 8: A photographer is taking a picture of a 4-foot long painting hung in an art gallery. The camera
lens is 1 foot below the lower edge of the painting, as shown in the figure provided. How far should the
camera be from the painting to maximize the angle subtended by the camera lens?
Related Rates…..They’re baaaack.
Example 9: A potrol car is parked 50 feet from a long warehouse (sse figure). The revolving light on top of
the car turns at a rate of 30 revolutions per minute.
a. Write  as a function of x.
b. How fast is the light beam moving along the wall when the beam makes an angle of   45 with the line
perpendicular from the light to the wall?