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Mastering Trigonometric Ratios and Functions

Learn to find exact values of trigonometric functions, solve right triangles, and apply inverse trigonometric functions to solve problems. Explore angles of elevation and depression.

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Mastering Trigonometric Ratios and Functions

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  1. trigonometric ratios • trigonometric functions • sine • cosine • tangent • cosecant • secant • cotangent • reciprocal function • inverse trigonometric function • inverse sine • inverse cosine • inverse tangent • angle of elevation • angle of depression • solve a right triangle Vocabulary

  2. Key Concept 1

  3. Find Values of Trigonometric Ratios Find the exact values of the six trigonometric functions of θ. The length of the side opposite θ is 33, the length of the side adjacent to θ is 56, and the length of the hypotenuse is 65. Example 1

  4. Find Values of Trigonometric Ratios Answer: Example 1

  5. A. B. C. D. Find the exact values of the six trigonometric functions of θ. Example 1

  6. If , find the exact values of the five remaining trigonometric functions for the acute angle . Because sin  = , label the opposite side 1 and the hypotenuse 3. Use One Trigonometric Value to Find Others Begin by drawing a right triangle and labeling one acute angle . Example 2

  7. By the Pythagorean Theorem, the length of the leg adjacent to Use One Trigonometric Value to Find Others Example 2

  8. Use One Trigonometric Value to Find Others Answer: Example 2

  9. Answer: Use One Trigonometric Value to Find Others Example 2

  10. If tan  = , find the exact values of the five remaining trigonometric functions for the acute angle . A. B. C. D. Example 2

  11. If tan  = , find the exact values of the five remaining trigonometric functions for the acute angle . A. B. C. D. Example 2

  12. Key Concept 3

  13. Find a Missing Side Length Find the value of x. Round to the nearest tenth, if necessary. Example 3

  14. CheckYou can check your answer by substituting x = 5.73 into . Simplify. x = 5.73 Find a Missing Side Length Therefore, x is about 5.7. Answer:about 5.7 Example 3

  15. Find a Missing Side Length SPORTS A competitor in a hiking competition must climb up the inclined course as shown to reach the finish line. Determine the distance in feet that the competitor must hike to reach the finish line. (Hint: 1 mile = 5280 feet.) Example 4

  16. Tangent function θ = 48°, opp = x, and adj = 5280 Multiply each side by 5280. Use a calculator. Find a Missing Side Length An acute angle measure and the adjacent side length are given, so the tangent function can be used to find the opposite side length. So, the competitor must hike about 5864 feet to reach the finish line. Answer: about 5864 ft Example 4

  17. WALKING Ernie is walking along the course x, as shown. Find the distance he must walk. A. 569.7 ft B. 228.0 ft C. 69.5 ft D. 8.5 ft Example 4

  18. WALKING Ernie is walking along the course x, as shown. Find the distance he must walk. A. 569.7 ft B. 228.0 ft C. 69.5 ft D. 8.5 ft Example 4

  19. Key Concept 5

  20. Find a Missing Angle Measure Use a trigonometric function to find the measure of θ. Round to the nearest degree, if necessary. Example 5

  21. Sine function opp = 12 and hyp = 15.7 ≈ 50° Definition of inverse sine Find a Missing Angle Measure Because the measures of the side opposite  and the hypotenuse are given, use the sine function. Answer: Example 5

  22. Sine function opp = 12 and hyp = 15.7 ≈ 50° Definition of inverse sine Find a Missing Angle Measure Because the measures of the side opposite  and the hypotenuse are given, use the sine function. Answer:about 50° Example 5

  23. Use a trigonometric function to find the measure of θ. Round to the nearest degree, if necessary. A. 32° B. 40° C. 50° D. 58° Example 5

  24. Use an Angle of Elevation SKIING The chair lift at a ski resort rises at an angle of 20.75° while traveling up the side of a mountain and attains a vertical height of 1200 feet when it reaches the top. How far does the chair lift travel up the side of the mountain? Example 6

  25. Sine function θ = 20.75o, opp = 1200, and hyp = d Multiply each side by d. Divide each side by sin 20.75o. Use a calculator. Use an Angle of Elevation Because the measure of an angle and the length of the opposite side are given in the problem, you can use the sine function to find d. Answer:about 3387 ft Example 6

  26. AIRPLANE A person on an airplane looks down at a point on the ground at an angle of depression of 15°.The plane is flying at an altitude of 10,000 feet. How far is the person from the point on the ground to the nearest foot? A. 2588 ft B. 10,353 ft C. 37,321 ft D. 38,637 ft Example 6

  27. AIRPLANE A person on an airplane looks down at a point on the ground at an angle of depression of 15°.The plane is flying at an altitude of 10,000 feet. How far is the person from the point on the ground to the nearest foot? A. 2588 ft B. 10,353 ft C. 37,321 ft D. 38,637 ft Example 6

  28. Use Two Angles of Elevation or Depression SIGHTSEEING A sightseer on vacation looks down into a deep canyon using binoculars. The angles of depression to the far bank and near bank of the river below are 61° and 63°, respectively. If the canyon is 1250 feet deep, how wide is the river? Example 7

  29. Use Two Angles of Elevation or Depression Draw a diagram to model this situation. Because the angle of elevation from a bank to the top of the canyon is congruent to the angle of depression from the canyon to that bank, you can label the angles of elevation as shown. Label the horizontal distance from the near bank to the base of the canyon as x and the width of the river as y. Example 7

  30. Substitute Subtract from each side. Use a calculator. Use Two Angles of Elevation or Depression Therefore, the river is about 56 feet wide. Answer:about 56 ft Example 7

  31. HIKING The angle of elevation from a hiker to the top of a mountain is 25o. After the hiker walks 1000 feet closer to the mountain the angle of elevation is 28o. How tall is the mountain? A. 3791 ft B. 4294 ft C. 7130 ft D. 8970 ft Example 7

  32. HIKING The angle of elevation from a hiker to the top of a mountain is 25o. After the hiker walks 1000 feet closer to the mountain the angle of elevation is 28o. How tall is the mountain? A. 3791 ft B. 4294 ft C. 7130 ft D. 8970 ft Example 7

  33. Substitute. Multiply. Use a calculator. Solve a Right Triangle A.Solve ΔFGH. Round side lengths to the nearest tenth and angle measures to the nearest degree. Find f and h using trigonometric functions. Example 8

  34. Substitute. Multiply. Use a calculator. Solve a Right Triangle Because the measures of two angles are given, H can be found by subtracting F from 90o. 41.4° + H = 90° Angles H and F are complementary. H ≈ 48.6° Subtract. Therefore, H ≈ 49°, f ≈ 18.5, and h ≈ 21.0. Answer:H≈ 49°, f ≈ 18.5, h ≈ 21.0 Example 8

  35. 4.1 Homework p. 227-229; 3, 9, 21, 27, 33, 35, 39, 43, 47, 57, 63

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