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Trigonometric Ratios: Finding Unknown Angles and Side Lengths in Right Triangles

Learn how to find missing angles and side lengths in right triangles using trigonometric functions such as sine, cosine, and tangent. Practice with examples and applications in real-world scenarios.

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Trigonometric Ratios: Finding Unknown Angles and Side Lengths in Right Triangles

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  1. Think About It Given the measure of one of the acute angles in a right triangle, find the measure of the other acute angle. 1. 45° 2. 60° 3. 24° 4. 38° 30° 45° 66° 52°

  2. Application Lab • Pythagorean Theorem activity 1

  3. Find the unknown length for each right triangle with legs a and b and hypotenuse c. 5. b = 12, c =13 6. a = 3, b = 3 a = 5

  4. Caution! Make sure that your graphing calculator is set to interpret angle values as degrees. Press . Check that Degree and not Radian is highlighted in the third row.

  5. Objectives Understand and use trigonometric relationships of acute angles in triangles. Determine side lengths of right triangles by using trigonometric functions.

  6. Vocabulary trigonometric function sine cosine tangent

  7. A trigonometric function is a function whose rule is given by a trigonometric ratio. A trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta θ is traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon θ.

  8. Example 1: Finding Trigonometric Ratios Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ =

  9. Check It Out! Example 2 Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ =

  10. Application Lab • Pythagorean Theorem activity: Walk though the Desert:, change to sine, cosine, tangent

  11. You will frequently need to determine the value of trigonometric ratios for 30°,60°, and 45° angles as you solve trigonometry problems. Recall from geometry that in a 30°-60°-90° triangle, the ration of the side lengths is 1: 3 :2, and that in a 45°-45°-90° triangle, the ratio of the side lengths is 1:1: 2.

  12. ° Substitute 30° for θ, x for opp, and 74 for hyp. Substitute for sin 30°. Example 2: Finding Side Lengths of Special Right Triangles Use a trigonometric function to find the value of x. The sine function relates the opposite leg and the hypotenuse. Multiply both sides by 74 to solve for x. x = 37

  13. ° ° Substitute 45 for θ, x for opp, and 20 for hyp. Substitute for sin 45°. Check It Out! Example 2 Use a trigonometric function to find the value of x. The sine function relates the opposite leg and the hypotenuse. Multiply both sides by 20 to solve for x.

  14. Example 3: Sports Application In a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp? Substitute 15.1° for θ, h for opp., and 19 for hyp. Multiply both sides by 19. Use a calculator to simplify. 5 ≈ h The height above the water is about 5 ft.

  15. Check It Out! Example 3 A skateboard ramp will have a height of 12 in., and the angle between the ramp and the ground will be 17°. To the nearest inch, what will be the length l of the ramp? Substitute 17° for θ, lfor hyp., and 12 for opp. Multiply both sides by l and divide by sin 17°. Use a calculator to simplify. l ≈ 41 The length of the ramp is about 41 in.

  16. Example 4: Geology Application A biologist whose eye level is 6 ft above the ground measures the angle of elevation to the top of a tree to be 38.7°. If the biologist is standing 180 ft from the tree’s base, what is the height of the tree to the nearest foot? Step 1 Draw and label a diagram to represent the information given in the problem.

  17. Example 4 Continued Step 2 Let x represent the height of the tree compared with the biologist’s eye level. Determine the value of x. Use the tangent function. Substitute 38.7 for θ, x for opp., and 180 for adj. 180(tan 38.7°) = x Multiply both sides by 180. 144 ≈ x Use a calculator to solve for x.

  18. Example 4 Continued Step 3 Determine the overall height of the tree. x + 6 = 144 + 6 = 150 The height of the tree is about 150 ft.

  19. 60.7° 120 ft Check It Out! Example 4 A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7°. If the surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot? Step 1 Draw and label a diagram to represent the information given in the problem.

  20. Check It Out! Example 4 Continued Step 2 Let x represent the height of the hill compared with the surveyor’s eye level. Determine the value of x. Use the tangent function. Substitute 60.7 for θ, x for opp., and 120 for adj. 120(tan 60.7°) = x Multiply both sides by 120. 214 ≈ x Use a calculator to solve for x.

  21. Check It Out! Example 4 Continued Step 3 Determine the overall height of the roller coaster hill. x + 6 = 214 + 6 = 220 The height of the hill is about 220 ft.

  22. trigonometric function • sine • cosine • tangent • cosecants • secant • cotangent Vocabulary

  23. Helpful Hint In each reciprocal pair of trigonometric functions, there is exactly one “co”

  24. The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.

  25. 70 θ 24 Example 5: Finding All Trigonometric Functions Find the values of the six trigonometric functions for θ. Step 1 Find the length of the hypotenuse. Pythagorean Theorem. a2 + b2 = c2 Substitute 24 for a and 70 for b. c2 = 242 + 702 c2 = 5476 Simplify. Solve for c. Eliminate the negative solution. c= 74

  26. Example 5 Continued Step 2 Find the function values.

  27. 80 θ 18 Check It Out! Example 5 Find the values of the six trigonometric functions for θ. Step 1 Find the length of the hypotenuse. Pythagorean Theorem. a2 + b2 = c2 Substitute 18 for a and 80 for b. c2 = 182 + 802 c2 = 6724 Simplify. Solve for c. Eliminate the negative solution. c= 82

  28. Check It Out! Example 5 Continued Step 2 Find the function values.

  29. Lesson Quiz: Part I Solve each equation. Check your answer. 1. Find the values of the six trigonometric functions for θ.

  30. Lesson Quiz: Part II 2. Use a trigonometric function to find the value of x. 3. A helicopter’s altitude is 4500 ft, and a plane’s altitude is 12,000 ft. If the angle of depression from the plane to the helicopter is 27.6°, what is the distance between the two, to the nearest hundred feet? 16,200 ft

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