• How do I use
inverse sine, cosine,
and tangent
functions to find the
angles of a triangle?
5.4
Solve Right Triangles
Inverse Trigonometric Ratios
Let
B
A be an acute angle.
A
C
Inverse Tangent If tan A  x, then
1 BC
tan

m

A
1
tan x  mA.
AC
Inverse Sine
If sin A  y, then sin 1 BC  mA
1
sin y  mA.
AB
Inverse Cosine If cos A  z, then cos 1 AC  mA
1
cos z  mA.
AB
5.4
Solve Right Triangles
Example 1 Use an inverse tangent to find an angle measure
Use a calculator to approximate the
measure of A to the nearest tenth
of a degree.
C
16
B
16
4
0.8
Because tan A = _____ = _____ = _____,
20
5
1
0.8  mA. Using a calculator,
tan ___
20
A
1
38.65980825 __
... .
tan 0___
.8  __________
So, the measure of
38.7
A is approximately _______.
o
5.4
Solve Right Triangles
Checkpoint. Complete the following exercises.
1. In Example 1, use a calculator and an
inverse tangent to approximate C
to the nearest tenth of a degree.
20
tan C 
16
 1.25
tan 1.25  51.3
1
o
C
16
B
20
A
5.4
Solve Right Triangles
Example 2 Use an inverse sine and an inverse cosine
Let A and B be acute angles in two right triangles. Use a
calculator to approximate the measure of A and B to the
nearest tenth of a degree.
a. sin A = 0.76
Solution
1
a. mA  sin
________
0.76
o
 _____
49.5
b. cos B = 0.17
1
b. mB  cos
________
0.17
o
 _____
80.2
5.4
Solve Right Triangles
Example 3 Solve a right triangle
Solve the right triangle. Round
your answer to the nearest tenth.
B
40 ft
23o
67
o
C
A
Solution
Step 1 Find mB by using the Triangle Sum Theorem.
o
o
o
____
180  90  23  mB
o
____
67  mB
5.4
Solve Right Triangles
Example 3 Solve a right triangle
Solve the right triangle. Round
your answer to the nearest tenth.
B
40 ft
23o
A
Solution
Step 2 Approximate BC using a ______
sine ratio.
o BC
_______
sin 23 
40
o
__________
40  sin 23_  BC
__________
40 0.3907_  BC
____
15
.6  BC
67
o
15.6 ft
C
o
Write ratio for _________.
sin 23
Multiply each side by 40
___.
o
Approximate sin
________.
23
Simplify and round answer.
5.4
Solve Right Triangles
Example 3 Solve a right triangle
Solve the right triangle. Round
your answer to the nearest tenth.
B
40 ft
23o
A
36.8 ft
Solution
Step 3 Approximate AC using a ______
cosine ratio.
AC
_______
cos 23  40
o
__________
40 cos 23_  AC
o
__________
40 0.9205_  AC
____
36
.8  AC
67
o
15.6 ft
C
o
Write ratio for _________.
cos 23
Multiply each side by 40
___.
o
Approximate cos
________.
23
Simplify and round answer.
5.4
Solve Right Triangles
B
Example 3 Solve a right triangle
40 ft
Solve the right triangle. Round
your answer to the nearest tenth.
Solution
A
o
23o
36.8 ft
o
67
o
15.6 ft
C
o
The angle measures are _____,
23 _____,
67 and _____.
90
The side lengths are _____
40 feet, about _______
15.6 feet,
and about ______
36.8 feet.
5.4
Solve Right Triangles
Checkpoint. Complete the following exercises.
2. Find mT to the nearest tenth of a degree if
cos T = 0.64.
mT  cos 0.64  50.2
1
o
3. Find mD to the nearest tenth of a degree if sin
D = 0.48.
mD  sin 0.48  28.7
1
o
5.4
Solve Right Triangles
Checkpoint. Complete the following exercises.
4. Solve a right triangle that has a 50o angle and a
15 inch hypotenuse.
mB  180  90  50
B
mB  40
11.5 in
a
o
C
15sin 50  15
15
o
a  sin 50 15  11.5 in
b
o
15cos 50  15
o 15
b  cos 50 15  9.6 in
o
40 o
15 in
50 o
9.6 in
A
5.4
Solve Right Triangles
Pg. 187, 5.4 #1-30