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Lesson 13.1 , For use with pages 852-858

c = 10. ANSWER. a = 51. ANSWER. Lesson 13.1 , For use with pages 852-858. In right triangle ABC , a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing length. Give exact values. 1. a = 6, b = 8. 2. c = 10, b = 7. 2.5 km.

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Lesson 13.1 , For use with pages 852-858

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  1. c = 10 ANSWER a = 51 ANSWER Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing length. Give exact values. 1.a = 6, b = 8 2.c = 10, b = 7

  2. 2.5 km ANSWER Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing length. Give exact values. 3. If you walk 2.0 kilometers due east and than 1.5kilometersdue north, how far will you be from your starting point?

  3. Trigonometry and Angles 13.1

  4. Evaluate the six trigonometric functions of the angle θ. From the Pythagorean theorem, the length of the hypotenuse is √ 169 13 √ 52 + 122 = = opp hyp 13 12 sinθ cscθ = = = = 12 hyp opp 13 EXAMPLE 1 Evaluate trigonometric functions SOLUTION

  5. 5 13 adj hyp cosθ secθ = = = = 13 5 hyp adj 12 5 opp adj tanθ cotθ = = = = 5 12 adj opp EXAMPLE 1 Evaluate trigonometric functions

  6. Draw: a right triangle with acute angle θ such that the leg opposite θ has length 4 and the hypotenuse has length 7. By the Pythagorean theorem, the length xof the other leg is x = 33. √ 72 – 42 √ = EXAMPLE 2 Standardized Test Practice SOLUTION STEP 1

  7. opp 4 33 4 tanθ = = = adj √ 33 33 ANSWER The correct answer is B. √ EXAMPLE 2 Standardized Test Practice Find the value of tan θ. STEP 2

  8. 1. From the Pythagorean theorem, the length of the hypotenuse is √ 25 5 32 + 42 = = opp hyp 3 5 √ sinθ cscθ = = = = hyp opp 5 3 for Examples 1 and 2 GUIDED PRACTICE Evaluate the six trigonometric functions of the angle θ. SOLUTION

  9. adj hyp 4 5 cosθ secθ = = = = 5 4 hyp adj opp adj 3 4 tanθ cotθ = = = = adj opp 4 3 for Examples 1 and 2 GUIDED PRACTICE

  10. From the Pythagorean theorem, the length of the adjacent is √ 64 8. 172 – 152 = = opp hyp 15 17 √ sinθ cscθ = = = = hyp opp 17 15 for Examples 1 and 2 GUIDED PRACTICE Evaluate the six trigonometric functions of the angle θ. SOLUTION

  11. adj hyp 8 17 cosθ secθ = = = = 17 8 hyp adj opp adj 15 8 tanθ cotθ = = = = adj opp 8 15 for Examples 1 and 2 GUIDED PRACTICE

  12. From the Pythagorean theorem, the length of the adjacent is 25 5 √ (5 22– 52 = √ √ = opp hyp 5 √ 5 2 sinθ cscθ = = = = hyp opp 5 √ 5 2 for Examples 1 and 2 GUIDED PRACTICE Evaluate the six trigonometric functions of the angle θ. SOLUTION

  13. adj hyp 5 √ 5 2 cosθ secθ = = = = hyp adj 5 √ 5 2 opp adj 5 5 tanθ cotθ = = = = adj opp 5 5 for Examples 1 and 2 GUIDED PRACTICE = 1 = 1

  14. Find the value of x for the right triangle shown. adj cos30º = hyp √ 3 x = 8 2 EXAMPLE 3 Find an unknown side length of a right triangle SOLUTION Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x. Write trigonometric equation. Substitute.

  15. x √ 4 3 = ANSWER √ 4 3 The length of the side is x= 6.93. EXAMPLE 3 Find an unknown side length of a right triangle Multiply each side by 8.

  16. Solve ABC. A and B are complementary angles, so B= 90º – 28º opp hyp tan28° sec28º = = adj adj a c tan28º sec28º = = 15 15 EXAMPLE 4 Use a calculator to solve a right triangle SOLUTION = 68º. Write trigonometric equation. Substitute.

  17. a c 7.98 17.0 So,B = 62º, a 7.98, and c 17.0 ANSWER ) ( 1 15 = c cos28º EXAMPLE 4 Use a calculator to solve a right triangle Solve for the variable. 15(tan28º) = a Use a calculator.

  18. Solve ABCusing the diagram at the right and the given measurements. 5. B = 45°, c = 5 A and B are complementary angles, so A= 90º – 45º adj opp cos45° sin45º = = hyp hyp a b cos45º sin45º = = 5 5 for Examples 3 and 4 GUIDED PRACTICE SOLUTION = 45º. Write trigonometric equation. Substitute.

  19. 5(sin45º) = b a b 3.54 3.54 So,A = 45º, b 3.54, and a 3.54. ANSWER for Examples 3 and 4 GUIDED PRACTICE Solve for the variable. 5(cos45º) = a Use a calculator.

  20. 6. A = 32°, b = 10 A and B are complementary angles, so B= 90º – 32º opp hyp tan32° sec32º = = adj adj a c tan32º sec32º = = 10 10 for Examples 3 and 4 GUIDED PRACTICE SOLUTION = 58º. Write trigonometric equation. Substitute.

  21. a c 6.25 11.8 So,B = 58º, a 6.25, and c 11.8. ANSWER ) ( 1 10 = c cos32º for Examples 3 and 4 GUIDED PRACTICE Solve for the variable. 10(tan32º) = a Use a calculator.

  22. 7. A = 71°, c = 20 A and B are complementary angles, so B= 90º – 71º adj opp cos71° sin71º = = hyp hyp b a cos71º sin71º = = 20 20 for Examples 3 and 4 GUIDED PRACTICE SOLUTION = 19º. Write trigonometric equation. Substitute.

  23. b a 6.51 18.9 So,B = 19º, b 6.51, and a 18.9. ANSWER for Examples 3 and 4 GUIDED PRACTICE Solve for the variable. 20(cos71º) = b 20(sin71º) = a Use a calculator.

  24. 8.B = 60°, a = 7 A and B are complementary angles, so A= 90º – 60º hyp opp sec60° tan60º = = adj adj b c sec60º tan60º = = 7 7 for Examples 3 and 4 GUIDED PRACTICE SOLUTION = 30º. Write trigonometric equation. Substitute.

  25. ) ( 1 7 = c cos60º b 12.1 ANSWER So,A = 30º, c =14, and b 12.1. for Examples 3 and 4 GUIDED PRACTICE 7(tan60º) = b Solve for the variable. 14 = c Use a calculator.

  26. While standing at Yavapai Point near the Grand Canyon, you measure an angle of 90º between Powell Point and Widforss Point, as shown. You then walk to Powell Point and measure an angle of 76º between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is about 2 miles. How wide is the Grand Canyon between Yavapai Point and Widforss Point? EXAMPLE 5 Use indirect measurement Grand Canyon

  27. x tan76º = 2 x 8.0 ≈ ANSWER The width is about 8.0 miles. EXAMPLE 5 Use indirect measurement SOLUTION Write trigonometric equation. 2(tan76º) = x Multiply each side by 2. Use a calculator.

  28. A parasailer is attached to a boat with a rope 300 feetlong. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat. EXAMPLE 6 Use an angle of elevation Parasailing

  29. ANSWER The height of the parasailer above the boat is about 223 feet. h sin48º = 300 x 223 ≈ EXAMPLE 6 Use an angle of elevation SOLUTION STEP 1 Draw: a diagram that represents the situation. STEP 2 Write: and solve an equation to find the height h. Write trigonometric equation. 300(sin48º) = h Multiply each side by 300. Use a calculator.

  30. 9. In Example 5, find the distance between Powell Point and Widforss Point. x sec76º = 2 ) ( 1 2 = x cos76º Substitute for sec 76°. 1 x 8.27 ≈ cos 76° The distance is about 8.27 miles. ANSWER for Examples 5 and 6 GUIDED PRACTICE Grand Canyon SOLUTION Write trigonometric equation. 2 sec76º = x Multiply each side by 2. Use a calculator.

  31. What If? In Example 6, estimate the height of the parasailer above the boat if the angle of elevation is 38°. 10. ANSWER The height of the parasailer above the boat is about 185feet. h sin38º = 300 h ≈ 185 for Examples 5 and 6 GUIDED PRACTICE SOLUTION Write trigonometric equation. 300(sin38º) = h Multiply each side by 300. Use a calculator.

  32. While standing at Yavapai Point near the Grand Canyon, you measure an angle of 90º between Powell Point and Widforss Point, as shown. You then walk to Powell Point and measure an angle of 76º between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is about 2 miles. How wide is the Grand Canyon between Yavapai Point and Widforss Point? EXAMPLE 5 Use indirect measurement Grand Canyon

  33. x tan76º = 2 x 8.0 ≈ ANSWER The width is about 8.0 miles. EXAMPLE 5 Use indirect measurement SOLUTION Write trigonometric equation. 2(tan76º) = x Multiply each side by 2. Use a calculator.

  34. A parasailer is attached to a boat with a rope 300 feetlong. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat. EXAMPLE 6 Use an angle of elevation Parasailing

  35. ANSWER The height of the parasailer above the boat is about 223 feet. h sin48º = 300 x 223 ≈ EXAMPLE 6 Use an angle of elevation SOLUTION STEP 1 Draw: a diagram that represents the situation. STEP 2 Write: and solve an equation to find the height h. Write trigonometric equation. 300(sin48º) = h Multiply each side by 300. Use a calculator.

  36. 9. In Example 5, find the distance between Powell Point and Widforss Point. x sec76º = 2 ) ( 1 2 = x cos76º Substitute for sec 76°. 1 x 8.27 ≈ cos 76° The distance is about 8.27 miles. ANSWER for Examples 5 and 6 GUIDED PRACTICE Grand Canyon SOLUTION Write trigonometric equation. 2 sec76º = x Multiply each side by 2. Use a calculator.

  37. What If? In Example 6, estimate the height of the parasailer above the boat if the angle of elevation is 38°. 10. ANSWER The height of the parasailer above the boat is about 185feet. h sin38º = 300 h ≈ 185 for Examples 5 and 6 GUIDED PRACTICE SOLUTION Write trigonometric equation. 300(sin38º) = h Multiply each side by 300. Use a calculator.

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