1. Use this diagram for Exercises 1–4.
1. If PR = 12 and m R = 19°, find p.
ANSWER
11.3
2. If m P = 58° and r = 5, find p.
ANSWER
8.0
3. If m P = 60°, and p = 9 , find q.
10.4
4. If r = 8 and p = 12, find q.
14.4

2. Solve Right Triangles
5.4 (M2)

3. Vocabulary
“To solve a right triangle” means to find the measures of all of its sides and angles.
Inverse Tangent: C B A

4. C B A
Inverse Trigonometry

5. 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.
SOLUTION
Because tan A =
1520 34 =
= 0.75,
tan–1 0.75
= m A
Use a calculator.
tan –
ANSWER
So, the measure of A is approximately 36.9o.

6. 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 –
81.4o

7. 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

8. 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 B
48o
= m B

9. Approximate BC by using a tangent ratio.
EXAMPLE 3
Solve a right triangle
STEP 2
Approximate BC by using a tangent ratio.
tan 42o =
BC70
Write ratio for tangent of 42o.
70 tan 42o
= BC
Multiply each side by 70. BC
Approximate tan 42o 63 BC
Simplify and round answer.

10. Approximate AB by using a cosine ratio.
EXAMPLE 3
Solve a right triangle
STEP 3
Approximate AB by using a cosine ratio.
cos 42o =
70 AB
Write ratio for cosine of 42o.
AB cos 42o =
Multiply each side by AB. AB
cos 42o =
Divide each side by cos 42o. AB
Use a calculator to find cos 42o. AB
94.2
Simplify .
ANSWER
The angle measures are 42o, 48o, and 90o. The
side lengths are 70 feet, about 63 feet, and about 94 feet.

11. 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?

12. 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. hyp
2 30
0.0667 x
sin –
ANSWER
The rake is about 3.8o, so it is within the suggested range of 5o or less.

13. 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.