Download 1.  2.  SOLUTION:  

When t =
4-6 Inverse Trigonometric Functions
1. sin
= . Therefore, arcsin
.
Find the exact value of each expression, if it
exists.
–1
, sin t =
3. arcsin
0
SOLUTION: SOLUTION: Find a point on the unit circle on the interval
Find a point on the unit circle on the interval
with a y-coordinate of
with a y-coordinate of 0.
When t = 0, sin t = 0. Therefore, sin
–1
When t =
0 = 0.
, sin t =
.
. Therefore, arcsin =
.
2. arcsin
4. sin – 1
SOLUTION: Find a point on the unit circle on the interval
SOLUTION: with a y-coordinate of
.
Find a point on the unit circle on the interval
with a y-coordinate of
.
When t =
, sin t =
. Therefore, arcsin
= When t =
.
, sin t =
. Therefore, sin
–1 = .
5. 3. arcsin
SOLUTION: SOLUTION: Find a point on the unit circle on the interval
Find a point on the unit circle on the interval
with a y-coordinate of
eSolutions Manual - Powered by Cognero
.
with a y-coordinate of
.
Page 1
–1 4-6 When
Inverse
Functions
t = Trigonometric
, sin t = . Therefore,
sin
= When t =
.
, cos t = 0. Therefore, arccos 0 =
.
7. cos– 1
5. SOLUTION: SOLUTION: Find a point on the unit circle on the interval
Find a point on the unit circle on the interval
with a y-coordinate of
When t =
=
, sin t =
with an x-coordinate of
.
. Therefore, sin
–1
When t =
, cos t =
.
–1
.
.
8. arccos (–1)
6. arccos 0
SOLUTION: SOLUTION: Find a point on the unit circle on the interval
with an x-coordinate of 0.
Find a point on the unit circle on the interval
with an x-coordinate of –1.
When t =
=
. Therefore, cos
, cos t = 0. Therefore, arccos 0 =
When t =
.
, cos t = –1. Therefore, arccos (–1)=
.
9. –1
7. cos
SOLUTION: Find a point on the unit circle on the interval
SOLUTION: Find a point on the unit circle on the interval
with an x-coordinate of
eSolutions Manual - Powered by Cognero
with an x-coordinate of
.
.
Page 2
4-6 Inverse Trigonometric Functions
When t =
, cos t = –1. Therefore, arccos (–1)=
.
When t =
, cos t =
–1
. Therefore, cos
=
.
15. ARCHITECTURE The support for a roof is 9. shaped like two right triangles, as shown below. Find
θ.
SOLUTION: Find a point on the unit circle on the interval
with an x-coordinate of
.
SOLUTION: Use inverse trigonometric functions and the unit
circle to solve.
Find a point on the unit circle on the interval
with a y-coordinate of
When t =
, cos t =
. Therefore,
.
=
.
10. cos– 1
SOLUTION: Find a point on the unit circle on the interval
with an x-coordinate of
When t =
, sin t =
. Therefore, sin
–1
= .
.
16. RESCUE A cruise ship sailed due west 24 miles before turning south. When the cruise ship became
disabled and the crew radioed for help, the rescue
boat found that the fastest route covered a distance
of 48 miles. Find the angle θ at which the rescue
boat should travel to aid the cruise ship.
When t =
, cos t =
–1
. Therefore, cos
=
.
15. ARCHITECTURE The support for a roof is shaped like two right triangles, as shown below. Find
θ.
SOLUTION: Use inverse trigonometric functions and the unit
circle to solve.
Find a point on the unit circle on the interval
with an x-coordinate of
SOLUTION: eSolutions
Manual - Powered by Cognero
Use inverse trigonometric functions and the unit
circle to solve.
.
Page 3
When t =
–1
4-6 When
Inverse
Functions
t = Trigonometric
, sin t = . Therefore,
sin
= .
16. RESCUE A cruise ship sailed due west 24 miles before turning south. When the cruise ship became
disabled and the crew radioed for help, the rescue
boat found that the fastest route covered a distance
of 48 miles. Find the angle θ at which the rescue
boat should travel to aid the cruise ship.
, cos t =
–1
. Therefore, cos
= .
Sketch the graph of each function.
17. y = arcsin x
SOLUTION: First, rewrite y = arcsin x in the form sin y = x.
to Next, assign values to y on the interval
make a table of values.
y
x = sin y
–1
SOLUTION: Use inverse trigonometric functions and the unit
circle to solve.
Find a point on the unit circle on the interval
with an x-coordinate of
0
0
1
.
Plot the points and connect them with a smooth
curve.
21. y = arccos x
When t =
, cos t =
–1
. Therefore, cos
= SOLUTION: First, rewrite y = arccos x in the form cos y = x.
.
Sketch the graph of each function.
17. y = arcsin x
Next, assign values to y on the interval
make a table of values.
x = cos y
y
SOLUTION: First, rewrite y = arcsin x in the form sin y = x.
0
to 1
Next, assign values to y on the interval
to 0
make a table of values.
eSolutions Manual - Powered by Cognero
y
-1
x = sin y
–1
Plot the points and connect them with a smooth
Page 4
lies on the interval [–1, 1]. Therefore,
=
4-6 Inverse Trigonometric Functions
The inverse property applies, because
21. y = arccos x
.
30. SOLUTION: First, rewrite y = arccos x in the form cos y = x.
SOLUTION: The inverse property applies, because
lies on the interval [–1, 1]. Therefore,
=
to Next, assign values to y on the interval
make a table of values.
x = cos y
y
0
.
1
0
-1
Plot the points and connect them with a smooth
curve.
Find the exact value of each expression, if it
exists.
29. SOLUTION: The inverse property applies, because
lies on the interval [–1, 1]. Therefore,
=
.
30. SOLUTION: TheManual
inverse
property
eSolutions
- Powered
by applies,
Cognero
interval [–1, 1]. Therefore,
because
lies on the =
.
Page 5