Warm-Up Up Exercises. Use this diagram for Exercises If PR = 12 and m R = 19, find p. ANSWER If m P = 58 and r = 5, find p.

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1 Warm-Up Up Exercises Use this diagram for Exercises If PR = 12 and m R = 19, find p. ANSWER If m P = 58 and r = 5, find p. ANSWER 8.0

2 Warm-Up Up Exercises Use this diagram for Exercises If m P = 60, and p = 9, find q. ANSWER If r = 8 and p = 12, find q. ANSWER 14.4

3 7.7 Solve Right Triangles Goal: Use inverse tangent, sine, and cosine ratios. Key Vocabulary: solve a right triangle inverse sine inverse cosine inverse tangent

4 7.7Warm-Up Solve Exercises Right Triangles Goal: Use inverse tangent, sine, and cosine ratios. Postulates, Corollaries, and Theorems: NONE

5 Definitions SOLVE A RIGHT TRIANGLE: Too find the measures of all sides of a right triangle.

6 Definitions INVERSE TANGENT: An inverse trigonometric ratio, abbreviated as tan -1. For an acute angle A, if tan A = x, then

7 Definitions INVERSE SINE: An inverse trigonometric ratio, abbreviated as sin -1. For an acute angle A, if sin A = y, then

8 Definitions INVERSE COSINE: An inverse trigonometric ratio, abbreviated as cos -1. For an acute angle A, if cos A = z, then

9 EXAMPLE 1 Use an inverse tangent to find an angle measure Use a calculator to approximate the measure of the nearest tenth of a degree. A to

10 EXAMPLE 1 Use an inverse tangent to find an angle measure Use a calculator to approximate the measure of the nearest tenth of a degree. A to SOLUTION Because tan A = Use a calculator. tan ANSWER 3 = = 0.75, tan = m A So, the measure of A is approximately 36.9 o.

11 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 and B to the nearest tenth of a degree. A a. sin A = 0.87 b. cos B = 0.15 SOLUTION a. m A = sin o b. m B = cos o

12 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.

13 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 53.1 o 2. Find m D to the nearest tenth of a degree if sin D = ANSWER 32.7 o

14 EXAMPLE 3 Solve a right triangle Solve the right triangle. Round decimal answers to the nearest tenth.

15 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. 180 o = 90 o + 42 o + m B 48 o = m B

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

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

18 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 5 o or less is generally preferred for the safety and comfort of the actors. Is the raked stage you are building within the range suggested?

19 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 x o = opp. hyp = x sin ANSWER The rake is about 3.8 o, so it is within the suggested range of 5 o or less.

20 GUIDED PRACTICE for Examples 3 and 4 3. Solve a right triangle that has a 40 o angle and a 20 inch hypotenuse. ANSWER 40 o, 50 o, and 90 o, about 12.9 in., about 15.3 in. and 20 in.

21 Daily Homework Quiz Use this diagram for exercises If x = 9 and z = 11 find m X to the nearest tenth of a degree. ANSWER 54.9

22 Daily Homework Quiz Use this diagram for exercises If y = 5 and z = 12 find m X to the nearest tenth of a degree. ANSWER 65.4

23 Daily Homework Quiz Use this diagram for exercises If m Y = 17.4 and z = 12 solve XYZ. ANSWER The angles are 17.4, 72.6, and 90; the sides are 12, about 3.6, and about 11.5.

24 Closing To solve a right triangle means to find the measures of all the angles and all the sides. If you know the measures of a side length and an acute angle, you can use trig ratios to find lengths of the other two sides. If you know the lengths of two sides, you can use an inverse trig ratio to find the measure of an angle and you can use the Pythagorean Theorem to find the length of the third side.

25 Closing You can use the sine, cosine, and tangent ratios to find the length of a side of a right triangle. You can use the inverse sine, inverse cosine, or inverse tangent ratio to find the measures of the angles.

26 Closing Use the Pythagorean Theorem if you have two sides of the triangle. Use a trigonometric ratio if you have an angle measure and a side length.

27 7.7 Warm-Up Homework Exercises Page 479: 1-33 odds