ANSWER 8.0

ANSWER 11.3

2. If m P = 58° and r = 5, find p.

1. If PR = 12 and m R = 19°, find p.

Use this diagram for Exercises 1–4.

4. If r = 8 and p = 12, find q.

14.4

10.4

3. If m P = 60°, and p = 9 , find q.

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:

C

B

A

1tanBC

m AAC

1tan , tanIf A x then x m A

C

B

A

1sinBC

m AAB

1sin , sinIf A y then y m A

1cosAC

m AAB

1cos , cosIf A z then z m A

Inverse Trigonometry

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 –1 0.75 36.86989765. . .

ANSWER

So, the measure of A is approximately 36.9o.

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 B48o = m B

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.

Approximate tan 42o70 0.9004 BC

63 BC Simplify and round answer.

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 = 70 Multiply each side by AB.

Divide each side by cos 42o.

Use a calculator to find cos 42o.

AB 70 cos 42o=

AB 70 0.7431

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.

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. hyp

= 2 30

0.0667

x sin –1 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.