9.1 Use Trigonometry with Right Triangles

1. 9.1 Use Trigonometry with Right Triangles • Use the Pythagorean Theorem to find missing lengths in right triangles. • Find trigonometric ratios using right triangles. • Use trigonometric ratios to find angle measures in right triangles. • Use special right triangles to find lengths in right triangles.

2. c a Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. a2 + b2 = c2 b

3. c = 10 ANSWER a = 51 ANSWER 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

4. Definition Trigonometry—the study of the special relationships between the angle measures and the side lengths of right triangles. A trigonometric ratio is the ratio the lengths of two sides of a right triangle.

5. B A C Right triangle parts. hypotenuse leg leg Name the hypotenuse. Name the legs. The Greek letter theta

6. Opposite or Adjacent? west • The opposite of east is ______ • The door is opposite the windows. • In a ROY G BIV, red is adjacent to _______ • The door is adjacent to the cabinet. orange

7. B A C Triangle Parts Opposite side hypotenuse Adjacent side c Looking from angle B: Which side is the opposite? Which side is the adjacent? a b a b Opposite side Adjacent side Looking from angle A: Which side is the hypotenuse? Which side is the opposite? Which side is the adjacent? c a b

8. B c a b A C Trigonometric ratios =

9. B c a b A C Trigonometric ratios and definition abbreviations SOHCAHTOA

10. B c a b A C Trigonometric ratios and definition abbreviations s

11. 1. adj hyp 4 5 cosθ secθ = = = = 5 4 hyp adj opp adj 3 4 tanθ cotθ = = = = adj opp 4 3 From the Pythagorean theorem, the length of the hypotenuse is √ 25 5. 32 + 42 = = opp hyp 3 5 √ sinθ cscθ = = = = hyp opp 5 3 Evaluate the six trigonometric functions of the angle θ. SOLUTION

12. adj hyp 5 √ 5 2 cosθ secθ = = = = hyp adj 5 √ 5 2 opp adj 5 5 tanθ cotθ = = = = adj opp 5 5 From the Pythagorean theorem, the length of the adjacent is opp hyp 5 √ 5 2 sinθ cscθ = = = = hyp opp 5 √ 5 2 Evaluate the six trigonometric functions of the angle θ. SOLUTION = 1 = 1

13. 45°- 45°- 90° Right Triangle In a 45°- 45°- 90° triangle, the hypotenuse is √2 times as long as either leg. The ratios of the side lengths can be written l-l-l√2. l l

14. 60° 2l l 30° 30°- 60° - 90° Right Triangle In a 30°- 60° - 90° triangle, the hypotenuse is twice as long as the shorter leg (the leg opposite the 30° angle, and the longer leg (opposite the 60° angle) is √3 times as long as the shorter leg. The ratios of the side lengths can be written l - l√3 – 2l.

15. In a right triangle, θ is an acute angle and cos θ= . What is sinθ? 4. opp 51 sinθ = = Draw: a right triangle with acute angle θ such that the leg opposite θ has length 7and the hypotenuse has length 10. By the Pythagorean theorem, the length xof the other leg is 10 hyp ANSWER 7 10 51 sinθ = 10 √ √ x √ = √ 51. 102 – 72 = SOLUTION STEP 1 Find: the value of sinθ. STEP 2

16. Find the value of x for the right triangle shown. x √ 4 3 = ANSWER √ 4 3 The length of the side is x= 6.93. adj cos30º = hyp √ 3 x = 8 2 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. Multiply each side by 8.

17. Solve ABC. a c 7.98 17.0 So,B = 62º, a 7.98, and c 17.0. A and B are complementary angles, so B= 90º – 28º ANSWER opp adj tan28° cos28º = = adj hyp a 15 tan28º cos28º = = 15 c Make sure your calculator is set to degrees. SOLUTION = 68º. Write trigonometric equation. Substitute. c Solve for the variable. 15(tan28º) = a Use a calculator.

18. Solve ABCusing the diagram at the right and the given measurements. 5(sin45º) = b a b 3.54 3.54 A and B are complementary angles, So,A = 45º, b 3.54, and a 3.54. 5. B = 45°, c = 5 ANSWER so A= 90º – 45º adj opp cos45° sin45º = = hyp hyp a b cos45º sin45º = = 5 5 SOLUTION = 45º. Write trigonometric equation. Substitute. Solve for the variable. 5(cos45º) = a Use a calculator.

19. 6. A = 32°, b = 10 a c 6.25 11.8 A and B are complementary angles, so B= 90º – 32º So,B = 58º, a 6.25, and c 11.8. opp hyp tan32° sec32º = = adj adj ANSWER a c tan32º sec32º = = 10 10 SOLUTION = 58º. Write trigonometric equation. Substitute. Solve for the variable. 10(tan32º) = a Use a calculator.

20. 7. A = 71°, c = 20 b a 6.51 18.9 A and B are complementary angles, So,B = 19º, b 6.51, and a 18.9. so B= 90º – 71º adj opp cos71° sin71º = = hyp hyp ANSWER b a cos71º sin71º = = 20 20 SOLUTION = 19º. Write trigonometric equation. Substitute. Solve for the variable. 20(cos71º) = b 20(sin71º) = a Use a calculator.

21. ) ( 1 8.B = 60°, a = 7 7 = c cos60º b 12.1 A and B are complementary angles, so A= 90º – 60º hyp opp sec60° tan60º = = adj adj So,A = 30º, c =14, and b 12.1. ANSWER b c sec60º tan60º = = 7 7 SOLUTION = 30º. Write trigonometric equation. Substitute. 7(tan60º) = b Solve for the variable. Use a calculator. 14 = c

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

23. Use the Pythagorean Theorem to find missing lengths in right triangles. • Find trigonometric ratios using right triangles. Since, cosine, tangent, cotangent, secant, cosecant • Use trigonometric ratios to find angle measures in right triangles. • Use special right triangles to find lengths in right triangles. In a 45°-45°-90° right triangle, use In a 30°-60°-90° right triangle, use

24. 9.1 Assignment Page 560, 4-14 even, 17, 19, 22-26 even, skip 8