7.6 apply the sine and cosine ratios
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  • 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. 7.6
  • 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. 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. 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. 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. 7.6 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. 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. 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. 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. 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. 7.6
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 7.6 Homework Pg 495-498 #19, 21, 23, 33, 36