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# 7.6 apply the sine and cosine ratios

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### 7.6 apply the sine and cosine ratios

1. 1. 7.6 Apply the Sine and Cosine Ratios 7.6 Bell Thinger Use this diagram for Exercises 1-4. 1. Name the hypotenuse. ANSWER XZ 2. Name the leg opposite X. ANSWER YZ 3. Name the leg adjacent to X. ANSWER XY 4. If XY = 17 and m X = 41 , find YZ. ANSWER 14.78
2. 2. 7.6
3. 3. 7.6 Example 1 Find sin S and sin R. Write each answer as a fraction and as a decimal rounded to four places. SOLUTION sin S = opp. S hyp = RT SR = 63 65 0.9692 sin R = opp. R hyp = ST SR = 16 65 0.2462
4. 4. 7.6 Guided Practice Find sin X and sin Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 1. ANSWER 8 or 0.4706, 15 or 0.8824 17 17
5. 5. 7.6 Guided Practice Find sin X and sin Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. 2. ANSWER 3 or 0.6, 5 4 or 0.8 5
6. 6. 7.6 Example 2 Find cos U and cos W. Write each answer as a fraction and as a decimal. SOLUTION cos U = adj. to U = hyp UV = UW 18 = 3 = 0.6000 30 5 W cos W = adj. to = hyp WV = UW 24 = 4 = 0.8000 30 5
7. 7. 7.6 Example 3 DOG RUN You want to string cable to make a dog run from two corners of a building, as shown in the diagram. Write and solve a proportion using a trigonometric ratio to approximate the length of cable you will need.
8. 8. 7.6 Example 3 SOLUTION sin 35 = opp. hyp. Write ratio for sine of 35o. sin 35° = 11 x Substitute. x sin 35° = 11 x = x x Multiply each side by x. 11 sin 35° 11 0.5736 Divide each side by sin 35o. 19.2 Simplify. Use a calculator to find sin 35o. You will need a little more than 19 feet of cable.
9. 9. 7.6 Guided Practice In Exercises 3 and 4, find cos R and cos S. Write each answer as a decimal. Round to four decimal places, if necessary. 3. ANSWER 0.6, 0.8
10. 10. 7.6 Guided Practice In Exercises 3 and 4, find cos R and cos S. Write each answer as a decimal. Round to four decimal places, if necessary. 4. ANSWER 0.8824, 0.4706
11. 11. 7.6 Guided Practice 5. In Example 3, use the cosine ratio to find the length of the other leg of the triangle formed. ANSWER about 15.7 ft
12. 12. 7.6
13. 13. 7.6 Example 4 SKIING You are skiing on a mountain with an altitude of 1200 meters. The angle of depression is 21 . About how far do you ski down the mountain?
14. 14. 7.6 Example 4 SOLUTION sin 21 = opp. hyp. Write ratio for sine of 21o. sin 21° = 1200 x Substitute. x sin 21° = 1200 x = 1200 sin 21° Multiply each side by x. Divide each side by sin 21o x 1200 0.3584 Use a calculator to find sin 21o x 3348.2 Simplify. You ski about 3348 meters down the mountain.
15. 15. 7.6 Example 5 SKATEBOARD RAMP You want to build a skateboard ramp with a length of 14 feet and an angle of elevation of 26°. You need to find the height and length of the base of the ramp.
16. 16. 7.6 Example 5 SOLUTION STEP 1 Find the height. opp. sin 26 = Write ratio for sine of 26o. hyp. sin 26° = x Substitute. 14 Multiply each side by 14. 14 sin 26° = x Use a calculator to simplify. 6.1 x The height is about 6.1 feet. STEP 2 Find the length of the base. adj. cos 26° = Write ratio for cosine of 26o. hyp. cos 26° = y Substitute. 14 Multiply each side by 14. 14 cos 26° = y Use a calculator to simplify. 12.6 y The length of the base is about 12.6 feet.
17. 17. 7.6 Example 6 Use a special right triangle to find the sine and cosine of a 60o angle. SOLUTION Use the 30 - 60° - 90° Triangle Theorem to draw a right triangle with side lengths of 1, 3 , and 2. Then set up sine and cosine ratios for the 60° angle. 3 opp. sin 60° = = hyp. 2 adj. = hyp. 1 2 cos 60° = 0.08660 = 0.5000
18. 18. 7.6 Guided Practice 7. Use a special right triangle to find the sine and cosine of a 30° angle. ANSWER 1 , 2 3 2
19. 19. Exit Slip 7.6 Use this diagram for Exercises 1- 3. 1. If x = 4 5 , y = 4 and z = 4 6 , find sin X, sin Y, cos X and cos Y. ANSWER 30 6 cos X = 6 6 sin X = 0.9129, sin Y = 0.4082, cos Y = 6 6 30 6 0.4082, 0.9129
20. 20. Exit Slip 7.6 Use this diagram for Exercises 1- 3. 2. If y = 10 and m ANSWER 38.6 Y = 15 , find z to the nearest tenth.
21. 21. Exit Slip 7.6 Use this diagram for Exercises 1- 3. 3. If z = 12 and m ANSWER 1.3 X = 84 , find y to the nearest tenth.
22. 22. Exit Slip 7.6 4. A lamppost is 11 feet tall. If the angle of elevation of the sun is 49 , how far is the top of the lamppost from the tip of its shadow? ANSWER 14.6 ft
23. 23. 7.6 Homework Pg 495-498 #19, 21, 23, 33, 36