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

2. Use trigonometric ratios to solve right triangles.
Target
Use trigonometric ratios to solve right triangles.
You will…
Use inverse tangent, sine, and cosine ratios.

3. Vocabulary
solve a right triangle – find the measure of all of its sides and angles; you can solve a right triangle if you know…
two side lengths or
one side length and one acute angle measure
inverse trigonometric ratios – are used to find angle measures
inverse tangent – If tan A = x, then tan-1 x = m A.
inverse sine – If sin A = y, then sin-1 y = m A.
inverse cosine – If cos A = z, then cos-1 z = m A.

4. 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
tan A =
1520 34 =
tan–1 34
tan-1(tan A) =
m A =
tan–1 34
Use a calculator.
tan –
ANSWER
So, the measure of A is approximately 36.9o.

5. 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
sin A = 0.87
cos B = 0.15
sin-1(sin A) =
sin –1 0.87
cos-1(cos B) =
cos –1 0.15
m A
= sin –1 0.87
m B
= cos –1 0.15
60.5o
m A
81.4o
m B

6. 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

7. 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 theTriangle Sum Theorem.
180o
= 90o + 42o + m B
48o
= m B

8. 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
70 tan 42o
= BC
Multiply each side by 70. BC
Approximate tan 42o 63 BC
Simplify and round answer.

9. 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
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.

10. 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?

11. 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.

12. 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.
WHAT IF?
In Example 4, suppose another raked stage is 20 feet long from front to back with a total rise of 2 feet. Is this raked stage safe? Explain.
No; the rake is 5.7° so it is slightly larger than the
suggested range.
ANSWER