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**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

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Solve Right Triangles 5.4 (M2)

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

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C B A Inverse Trigonometry

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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.

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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

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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

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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

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**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.

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**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.

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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?

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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.

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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.

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