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

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

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.

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 –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 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

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

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. 70 0.9004 BC Approximate tan 42o 63 BC Simplify and round answer.

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 = 70 Multiply each side by AB. AB 70 cos 42o = Divide each side by cos 42o. AB 70 0.7431 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.

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. 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