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Trigonometric Functions in Right Triangles

This lesson covers the evaluation of trigonometric functions, finding trigonometric ratios, calculating missing side lengths, and using inverse trigonometric ratios in right triangles.

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Trigonometric Functions in Right Triangles

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 11) CCSS Then/Now New Vocabulary Key Concept: Trigonometric Functions in Right Triangles Example 1: Evaluate Trigonometric Functions Example 2: Find Trigonometric Ratios Key Concept: Trigonometric Values for Special Angles Example 3: Find a Missing Side Length Example 4: Find a Missing Side Length Key Concept: Inverse Trigonometric Ratios Example 5: Find a Missing Angle Measure Example 6: Use Angles of Elevation and Depression Lesson Menu

  3. When a triangle is a right triangle, one of its angles measures 90°. Does this show correlation or causation? Explain. A. Causation; a triangle must have a 90° angle to be a right triangle. B. Causation; a triangle’s angles must add to 180°. C. Correlation; a triangle must have a 90° angle to be a right triangle. D. Correlation; a triangle’s angles must add to 180°. 5-Minute Check 1

  4. When a triangle is a right triangle, one of its angles measures 90°. Does this show correlation or causation? Explain. A. Causation; a triangle must have a 90° angle to be a right triangle. B. Causation; a triangle’s angles must add to 180°. C. Correlation; a triangle must have a 90° angle to be a right triangle. D. Correlation; a triangle’s angles must add to 180°. 5-Minute Check 1

  5. A. B. C. D. From a box containing 8 blue pencils and 6 red pencils, 4 pencils are drawn and not replaced. What is the probability that all four pencils are the same color? 5-Minute Check 2

  6. A. B. C. D. From a box containing 8 blue pencils and 6 red pencils, 4 pencils are drawn and not replaced. What is the probability that all four pencils are the same color? 5-Minute Check 2

  7. Test the null hypothesis for H0 = 82, h1 > 82, n = 150, x = 83.1, and  = 2.1. _ A. accept B. reject 5-Minute Check 3

  8. Test the null hypothesis for H0 = 82, h1 > 82, n = 150, x = 83.1, and  = 2.1. _ A. accept B. reject 5-Minute Check 3

  9. A. B. C. D. Jenny makes 60% of her foul shots. If she takes 5 shots in a game, what is the probability that she will make fewer than 4 foul shots? 5-Minute Check 4

  10. A. B. C. D. Jenny makes 60% of her foul shots. If she takes 5 shots in a game, what is the probability that she will make fewer than 4 foul shots? 5-Minute Check 4

  11. Mathematical Practices 6 Attend to precision. CCSS

  12. You used the Pythagorean Theorem to find side lengths of right triangles. • Find values of trigonometric functions for acute angles. • Use trigonometric functions to find side lengths and angle measures of right triangles. Then/Now

  13. trigonometry • trigonometric ratio • trigonometric function • sine • cosine • tangent • cosecant • secant • cotangent • reciprocal functions • inverse sine • inverse cosine • inverse tangent • angle of elevation • angle of depression Vocabulary

  14. Concept

  15. For this triangle, the leg opposite G is HF and the leg adjacent to G is GH. The hypotenuse is GF. Evaluate Trigonometric Functions Find the values of the six trigonometric functions for angle G. Use opp = 24, adj = 32, and hyp = 40 to write each trigonometric ratio. Example 1

  16. Evaluate Trigonometric Functions Example 1

  17. Evaluate Trigonometric Functions Answer: Example 1

  18. Answer: Evaluate Trigonometric Functions Example 1

  19. A. B. C.D. Find the value of the six trigonometric functions for angle A. Example 1

  20. A. B. C.D. Find the value of the six trigonometric functions for angle A. Example 1

  21. In a right triangle, A is acute and . Find the value of csc A. Step 1 Draw a right triangle and label one acute angle A. Since and , label the opposite leg 5 and the adjacent leg 3. Find Trigonometric Ratios Example 2

  22. Find Trigonometric Ratios Step 2 Use the Pythagorean Theorem to find c. a2 + b2 = c2 Pythagorean Theorem 32 + 52 = c2 Replace a with 3 and b with 5. 34 = c2 Simplify. Take the square root of each side. Length cannot be negative. Example 2

  23. Replace hyp with and opp with 5. Find Trigonometric Ratios Step 3 Now find csc A. Cosecant ratio Answer: Example 2

  24. Replace hyp with and opp with 5. Answer: Find Trigonometric Ratios Step 3 Now find csc A. Cosecant ratio Example 2

  25. A. B. C. D. Example 2

  26. A. B. C. D. Example 2

  27. Concept

  28. Find a Missing Side Length Use a trigonometric function to find the value of x. Round to the nearest tenth if necessary. The measure of the hypotenuse is 12. The side with the missing length is opposite the angle measuring 60. The trigonometric function relating the opposite side of a right triangle and the hypotenuse is the sine function. Example 3

  29. Find a Missing Side Length Sine ratio Replace  with 60°, opp with x, and hyp with 12. Multiply each side by 12. 10.4 ≈ x Use a calculator. Answer: Example 3

  30. x = Find a Missing Side Length Sine ratio Replace  with 60°, opp with x, and hyp with 12. Multiply each side by 12. 10.4 ≈ x Use a calculator. Answer: Example 3

  31. A. B. C. D. Write an equation involving sin, cos, or tan that can be used to find the value of x. Then solve the equation. Round to the nearest tenth. Example 3

  32. A. B. C. D. Write an equation involving sin, cos, or tan that can be used to find the value of x. Then solve the equation. Round to the nearest tenth. Example 3

  33. Find a Missing Side Length BUILDINGSTo calculate the height of a building, Joel walked 200 feet from the base of the building and used an inclinometer to measure the angle from his eye to the top of the building. If Joel’s eye level is at 6 feet, what is the distance from the top of the building to Joel’s eye? Example 4

  34. Replace  with 76°, adj with 200, and hyp with d. Solve for d. Find a Missing Side Length Cosine function Use a calculator. Answer: Example 4

  35. Replace  with 76°, adj with 200, and hyp with d. Solve for d. Find a Missing Side Length Cosine function Use a calculator. Answer: The distance from the top of the building to Joel’s eye is about 827 feet. Example 4

  36. TREES To calculate the height of a tree in his front yard, Anand walked 50 feet from the base of the tree and used an inclinometer to measure the angle from his eye to the top of the tree, which was 62°. If Anand’s eye level is at 6 feet, about how tall is the tree? A. 43 ft B. 81 ft C. 87 ft D. 100 ft Example 4

  37. TREES To calculate the height of a tree in his front yard, Anand walked 50 feet from the base of the tree and used an inclinometer to measure the angle from his eye to the top of the tree, which was 62°. If Anand’s eye level is at 6 feet, about how tall is the tree? A. 43 ft B. 81 ft C. 87 ft D. 100 ft Example 4

  38. Concept

  39. Find a Missing Angle Measure A. Find the measure of A. Round to the nearest tenth if necessary. You know the measures of the sides. You need to find mA. Inverse sine Example 5

  40. Find a Missing Angle Measure Use a calculator. Answer: Example 5

  41. Find a Missing Angle Measure Use a calculator. Answer: Therefore, mA≈ 32°. Example 5

  42. Inverse cosine Use a calculator. Find a Missing Angle Measure B. Find the measure of B. Round to the nearest tenth if necessary. Use the cosine function. Answer: Example 5

  43. Inverse cosine Use a calculator. Find a Missing Angle Measure B. Find the measure of B. Round to the nearest tenth if necessary. Use the cosine function. Answer: Therefore, mB ≈ 58º. Example 5

  44. A. Find the measure of A. A.mA = 72º B.mA = 80º C.mA = 30º D.mA = 55º Example 5

  45. A. Find the measure of A. A.mA = 72º B.mA = 80º C.mA = 30º D.mA = 55º Example 5

  46. B. Find the measure of B. A.mB = 18º B.mB = 10º C.mB = 60º D.mB = 35º Example 5

  47. B. Find the measure of B. A.mB = 18º B.mB = 10º C.mB = 60º D.mB = 35º Example 5

  48. Use Angles of Elevation and Depression A. GOLF A golfer is standing at the tee, looking up to the green on a hill. The tee is 36 yards lower than the green and the angle of elevation from the tee to the hole is 12°. From a camera in a blimp, the apparent distance between the golfer and the hole is the horizontal distance. Find the horizontal distance. Example 6

  49. tan  Use Angles of Elevation and Depression Write an equation using a trigonometric function that involves the ratio of the vertical rise (side opposite the 12° angle) and the horizontal distance from the tee to the hole (adjacent). Multiply each side by x. Divide each side by tan 12°. Simplify. x≈ 169.4 Answer: Example 6

  50. tan  Use Angles of Elevation and Depression Write an equation using a trigonometric function that involves the ratio of the vertical rise (side opposite the 12° angle) and the horizontal distance from the tee to the hole (adjacent). Multiply each side by x. Divide each side by tan 12°. Simplify. x≈ 169.4 Answer: So, the horizontal distance from the tee to the green as seen from a camera in a blimp is about 169.4 yards. Example 6

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