Use this diagram for Exercises 1–4.
1. If PR = 12 and m
ANSWER
2. If m
11.3
P = 58° and r = 5, find p.
ANSWER
3. If m
R = 19°, find p.
8.0
P = 60°, and p = 9 , find q.
10.4
4. If r = 8 and p = 12, find q.
14.4
Solve Right
Triangles
5.4 (M2)
Vocabulary


“To solve a right triangle” means to find the
measures of all of its sides and angles.
Inverse Tangent:
B
1
If tan A  x, then tan x  mA
C
tan
1
BC
 mA
AC
A
B
Inverse Trigonometry
C
If sin A  y, then sin 1 y  mA
BC
sin
 mA
AB
1
If cos A  z, then cos1 z  mA
AC
cos
 mA
AB
1
A
EXAMPLE 1
Use an inverse tangent to find an angle measure
Use a calculator to approximate the measure of
the nearest tenth of a degree.
A to
SOLUTION
15
3
Because tan A =
=
= 0.75, tan–1 0.75 = m
20
4
Use a calculator.
tan –1 0.75
36.86989765. . .
ANSWER
So, the measure of
A is approximately 36.9o.
A
EXAMPLE 2
Use an inverse sine and an inverse cosine
Let A and
B be acute angles in a right triangle.
Use a calculator to approximate the measures of A
and
B to the nearest tenth of a degree.
a. sin A = 0.87
b. cos B = 0.15
SOLUTION
a. m
A
= sin –1 0.87
60.5o
b. m
B
= cos –1 0.15
81.4o
GUIDED PRACTICE
for Examples 1 and 2
1. Look back at Example 1. Use a calculator and an
inverse tangent to approximate m
C to the
nearest tenth of a degree.
ANSWER
53.1o
2. Find m D to the nearest tenth of a degree if
sin D = 0.54.
ANSWER
32.7o
EXAMPLE 3
Solve a right triangle
Solve the right triangle. Round decimal answers to
the nearest tenth.
SOLUTION
STEP 1
Find m B by using the Triangle
Sum Theorem.
180o = 90o + 42o + m
48o = m B
B
EXAMPLE 3
Solve a right triangle
STEP 2
Approximate BC by using a tangent ratio.
tan 42o = BC
70
70 tan 42o = BC
Write ratio for tangent of 42o.
Multiply each side by 70.
70 0.9004
BC
Approximate tan 42o
63
BC
Simplify and round answer.
EXAMPLE 3
Solve a right triangle
STEP 3
Approximate AB by using a cosine ratio.
cos 42o = 70
Write ratio for cosine of 42o.
AB
AB cos 42o =
70
70
AB = cos 42o
70
AB
0.7431
AB
94.2
Multiply each side by AB.
Divide each side by cos 42o.
Use a calculator to find cos 42o.
Simplify .
ANSWER The angle measures are 42o, 48o, and 90o. The
EXAMPLE 4
Solve a real-world problem
THEATER DESIGN
Suppose your school is building a raked stage. The
stage will be 30 feet long from front to back, with a
total rise of 2 feet. A rake (angle of elevation) of 5o or
less is generally preferred for the safety and comfort
of the actors. Is the raked stage you are building
within the range suggested?
EXAMPLE 4
Solve a real-world problem
SOLUTION
Use the sine and inverse sine ratios to find the degree
measure x of the rake.
sin xo = opp. = 2
hyp
30
x
sin –1 0.0667
0.0667
3.842
ANSWER
The rake is about 3.8o, so it is within the suggested
range of 5o or less.
GUIDED PRACTICE
for Examples 3 and 4
3. Solve a right triangle that has a 40o angle and a 20
inch hypotenuse.
ANSWER
40o, 50o, and 90o, about 12.9 in., about 15.3 in. and 20 in.