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• 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 mA.
AC
Inverse Sine
If sin A y, then sin 1 BC mA
1
sin y mA.
AB
Inverse Cosine If cos A z, then cos 1 AC mA
1
cos z mA.
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 mA. 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. mA sin
________
0.76
o
_____
49.5
b. cos B = 0.17
1
b. mB 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 mB by using the Triangle Sum Theorem.
o
o
o
____
180 90 23 mB
o
____
67 mB
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 mT to the nearest tenth of a degree if
cos T = 0.64.
mT cos 0.64 50.2
1
o
3. Find mD to the nearest tenth of a degree if sin
D = 0.48.
mD 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.
mB 180 90 50
B
mB 40
11.5 in
a
o
C
15sin 50 15
15
o
a sin 50 15 11.5 in
b
o
15cos 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