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Section 4.5
Inverse Trigonometric
Functions
Objectives:
1. To define inverse trigonometric
functions.
2. To evaluate inverse trigonometric
functions.
3. To graph inverse trigonometric
functions.
Remember, to find the rule for an
inverse function, you interchange the
x and y and solve for y.
In the case of y = sin x, solving
x = sin y for y requires a symbol for
the inverse. Mathematicians use the
notation y = sin-1 x or y = arcsin x.
This notation means “y is the angle
whose sine is x.”
The notation sin-1 indicates an angle!
Consider the sin function. Since the
function is not one-to-one the inverse
is not a function.
1
-2
-
-1

2
To make the original function one-toone we restrict the domain of sin to
- 
[ , ]
2
2
1
-2
-
-1

2
This is the graph of f(x) = Sin x.
1
-2
-
-1

2
By reflecting Sin x across the line y =
x we get the graph of f(x) = Sin-1 x.
1
-2
-
-1

2
EXAMPLE 1 y = Sin-1 1. Find y.
y = Sin-1 1 is an equivalent expression to
sin y = 1. In other words, we want to know
the angle whose sin is 1. Since the “s” in
sin is capitalized we want the angle from
- 
the restricted domain [ , ].
2 2
EXAMPLE 1 y = Sin-1 1. Find y.
y = Sin-1 1

y=
2
EXAMPLE 2 Find
sin(Sin-1
1
 1
) = sin =
2
6 2
sin(Sin-1
1
)
2
3
EXAMPLE 3 Find
)
4
Cos-1 3/4 represents an angle in [0, ].
Since 3/4 is positive it is a first quadrant
angle. Therefore you have the following
right triangle.
sin(Cos-1
4

3
x
3
EXAMPLE 3 Find
)
4
Use the Pythagorean theorem to find the
missing side.
sin(Cos-1
3 2 + x2 = 4 2
9 + x2 = 16
x2 = 7
x= 7
4

3
x
EXAMPLE 3 Find
4

3
∴
sin(Cos-1
3
7
)=
4
4
sin(Cos-1
7
3
)
4
2
EXAMPLE 4 Find
)
2
Since the cosine is negative, the angle is

2
in the second quadrant. The cos = .
4
2
The angle in the second quadrant with a
2

reference angle of
is the angle  - =
2
4
3
.
4
Cos-1(-
Homework
pp. 196-197
►A. Exercises
Graph.
1. y = cos x
1

►A. Exercises
Graph.
2. y = Cos x, x  [0, ]
1

►A. Exercises
Graph.
3. y = Cos-1 x

1
1

►A. Exercises
Graph.
4. y = tan x
1

►A. Exercises
Graph.
- 
5. y = Tan x, x  ( 2 , 2 )
1

►A. Exercises
Graph.
6. y = Tan-1 x
►A. Exercises
Without using a calculator, find the
following values.
3
-1
7. Sin
2
►A. Exercises
Without using a calculator, find the
following values.
1
-1
13. tan(Sin )
2
►A. Exercises
Without using a calculator, find the
following values.
5
-1
15. cos(Sin
)
3
►A. Exercises
Use a calculator to determine the
following values.
17. Sin-1 0.3420
►B. Exercises
Graph the given function over its
appropriate restricted domain. (State the
restricted domain.) Graph its inverse
function on the same set of axes.
21. g(x) = Csc x
►B. Exercises
Use the definitions and a calculator to
evaluate the following.
23. Cot-1 0.684
►B. Exercises
Use the definitions and a calculator to
evaluate the following.
27. Sin-1 0.7854
■ Cumulative Review
35. Give the angle of inclination of the
line 3x + 4y = 7 to the nearest degree.
■ Cumulative Review
5
1
36. Change f(x) = x – to general
7
4
form.
■ Cumulative Review
37. Give the function rule for the line
passing through the points (-4, 5),
(3, 8.5), and (8, 11).
■ Cumulative Review
38. Find an equivalent expression for

f(x) = sec ( – x).
2
■ Cumulative Review
39. Find the inverse of the function
2
f(x) = x – 5.
3
y = sin x
1
-2
-
y = csc x
-1

2
y = cos x
1
-2
-
y = sec x
-1

2
y = tan x
1
-2
y = cot x
-
-1

2