1. Splash Screen

2. Five-Minute Check (over Lesson 8–3) CCSS Then/Now New Vocabulary
Key Concept: Trigonometric Ratios
Example 1: Find Sine, Cosine, and Tangent Ratios
Example 2: Use Special Right Triangles to Find Trigonometric Ratios
Example 3: Real-World Example: Estimate Measures Using Trigonometry
Key Concept: Inverse Trigonometric Ratios
Example 4: Find Angle Measures Using Inverse Trigonometric Ratios
Example 5: Solve a Right Triangle
Lesson Menu

3. Find x and y. A. B. C. D.
5-Minute Check 1

4. Find x and y. A. B. C. D.
5-Minute Check 1

5. Find x and y.
A. x = 5, y = 5
B. x = 5, y = 45 C. D.
5-Minute Check 2

6. Find x and y.
A. x = 5, y = 5
B. x = 5, y = 45 C. D.
5-Minute Check 2

7. The length of the diagonal of a square is centimeters
The length of the diagonal of a square is centimeters. Find the perimeter of the square.
A. 15 cm
B. 30 cm
C. 45 cm
D. 60 cm
5-Minute Check 3

8. The length of the diagonal of a square is centimeters
The length of the diagonal of a square is centimeters. Find the perimeter of the square.
A. 15 cm
B. 30 cm
C. 45 cm
D. 60 cm
5-Minute Check 3

9. The side of an equilateral triangle measures 21 inches
The side of an equilateral triangle measures 21 inches. Find the length of an altitude of the triangle.
A in.
B. 12 in.
C. 14 in.
D in.
5-Minute Check 4

10. The side of an equilateral triangle measures 21 inches
The side of an equilateral triangle measures 21 inches. Find the length of an altitude of the triangle.
A in.
B. 12 in.
C. 14 in.
D in.
5-Minute Check 4

11. ΔMNP is a 45°-45°-90° triangle with right angle P
ΔMNP is a 45°-45°-90° triangle with right angle P. Find the coordinates of M in Quadrant II for P(2, 3) and N(2, 8).
A. (–1, 3)
B. (–3, 3)
C. (5, 3)
D. (6, 2)
5-Minute Check 5

12. ΔMNP is a 45°-45°-90° triangle with right angle P
ΔMNP is a 45°-45°-90° triangle with right angle P. Find the coordinates of M in Quadrant II for P(2, 3) and N(2, 8).
A. (–1, 3)
B. (–3, 3)
C. (5, 3)
D. (6, 2)
5-Minute Check 5

13. The hypotenuse of a 30°-60°-90° triangle measures inches
The hypotenuse of a 30°-60°-90° triangle measures inches. What is the length of the side opposite the 30° angle?
A. 10 in.
B. 20 in. C. D.
5-Minute Check 6

14. The hypotenuse of a 30°-60°-90° triangle measures inches
The hypotenuse of a 30°-60°-90° triangle measures inches. What is the length of the side opposite the 30° angle?
A. 10 in.
B. 20 in. C. D.
5-Minute Check 6

15. Mathematical Practices
Content Standards
G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
5 Use appropriate tools strategically.
CCSS

16. Find trigonometric ratios using right triangles.
You used 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.
Then/Now

17. trigonometry trigonometric ratio sine cosine tangent inverse sine
inverse cosine
inverse tangent
Vocabulary

18. Concept

19. Find Sine, Cosine, and Tangent Ratios
A. Express sin L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

20. Find Sine, Cosine, and Tangent Ratios
A. Express sin L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

21. Find Sine, Cosine, and Tangent Ratios
B. Express cos L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

22. Find Sine, Cosine, and Tangent Ratios
B. Express cos L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

23. Find Sine, Cosine, and Tangent Ratios
C. Express tan L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

24. Find Sine, Cosine, and Tangent Ratios
C. Express tan L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

25. Find Sine, Cosine, and Tangent Ratios
D. Express sin N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

26. Find Sine, Cosine, and Tangent Ratios
D. Express sin N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

27. Find Sine, Cosine, and Tangent Ratios
E. Express cos N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

28. Find Sine, Cosine, and Tangent Ratios
E. Express cos N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

29. Find Sine, Cosine, and Tangent Ratios
F. Express tan N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

30. Find Sine, Cosine, and Tangent Ratios
F. Express tan N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

31. A. Find sin A. A. B. C. D.
Example 1

32. A. Find sin A. A. B. C. D.
Example 1

33. B. Find cos A. A. B. C. D.
Example 1

34. B. Find cos A. A. B. C. D.
Example 1

35. C. Find tan A. A. B. C. D.
Example 1

36. C. Find tan A. A. B. C. D.
Example 1

37. D. Find sin B. A. B. C. D.
Example 1

38. D. Find sin B. A. B. C. D.
Example 1

39. E. Find cos B. A. B. C. D.
Example 1

40. E. Find cos B. A. B. C. D.
Example 1

41. F. Find tan B. A. B. C. D.
Example 1

42. F. Find tan B. A. B. C. D.
Example 1

43. The side adjacent to the 60° angle has a measure of x.
Use Special Right Triangles to Find Trigonometric Ratios
Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth.
Draw and label the side lengths of a 30°-60°-90° right triangle, with x as the length of the shorter leg and 2x as the length of the hypotenuse.
The side adjacent to the 60° angle has a measure of x.
Example 2

44. Definition of cosine ratio
Use Special Right Triangles to Find Trigonometric Ratios
Definition of cosine ratio
Substitution
Simplify.
Example 2

45. Definition of cosine ratio
Use Special Right Triangles to Find Trigonometric Ratios
Definition of cosine ratio
Substitution
Simplify.
Example 2

46. Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D.
Example 2

47. Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D.
Example 2

48. Estimate Measures Using Trigonometry
EXERCISING A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor?
Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches.
Example 3

49. Use a calculator to find y.
Estimate Measures Using Trigonometry
Multiply each side by 60.
Use a calculator to find y.
KEYSTROKES:
ENTER
SIN
Answer:
Example 3

50. Use a calculator to find y.
Estimate Measures Using Trigonometry
Multiply each side by 60.
Use a calculator to find y.
KEYSTROKES:
ENTER
SIN
Answer: The treadmill is about 7.3 inches high.
Example 3

51. CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch?
A. 1 in.
B. 11 in.
C. 16 in.
D. 15 in.
Example 3

52. CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch?
A. 1 in.
B. 11 in.
C. 16 in.
D. 15 in.
Example 3

53. Concept

54. Use a calculator to find the measure of P to the nearest tenth.
Find Angle Measures Using Inverse Trigonometric Ratios
Use a calculator to find the measure of P to the nearest tenth.
Example 4

55. Find Angle Measures Using Inverse Trigonometric Ratios
The measures given are those of the leg adjacent to P and the hypotenuse, so write the equation using the cosine ratio.
KEYSTROKES: [COS]
2nd ( ÷ )
ENTER
Answer:
Example 4

56. Answer: So, the measure of P is approximately 46.8°.
Find Angle Measures Using Inverse Trigonometric Ratios
The measures given are those of the leg adjacent to P and the hypotenuse, so write the equation using the cosine ratio.
KEYSTROKES: [COS]
2nd ( ÷ )
ENTER
Answer: So, the measure of P is approximately 46.8°.
Example 4

57. Use a calculator to find the measure of D to the nearest tenth.
B. 48.3°
C. 55.4°
D. 57.2°
Example 4

58. Use a calculator to find the measure of D to the nearest tenth.
B. 48.3°
C. 55.4°
D. 57.2°
Example 4

59. Solve a Right Triangle
Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.
Example 5

60. Step 1 Find mA by using a tangent ratio.
Solve a Right Triangle
Step 1 Find mA by using a tangent ratio.
Definition of inverse tangent
≈ mA Use a calculator.
So, the measure of A is about 30.
Example 5

61. Step 2 Find mB using complementary angles.
Solve a Right Triangle
Step 2 Find mB using complementary angles.
mA + mB = 90 Definition of complementary angles
30 + mB ≈ 90 mA ≈ 30
mB ≈ 60 Subtract 30 from each side.
So, the measure of B is about 60.
Example 5

62. Step 3 Find AB by using the Pythagorean Theorem.
Solve a Right Triangle
Step 3 Find AB by using the Pythagorean Theorem.
(AC)2 + (BC)2 = (AB)2 Pythagorean Theorem
= (AB)2 Substitution
65 = (AB)2 Simplify.
Take the positive square root of each side.
8.06 ≈ AB Use a calculator.
Example 5

63. So, the measure of AB is about 8.06.
Solve a Right Triangle
So, the measure of AB is about 8.06.
Answer:
Example 5

64. So, the measure of AB is about 8.06.
Solve a Right Triangle
So, the measure of AB is about 8.06.
Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06
Example 5

65. Solve the right triangle
Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.
A. mA = 36°, mB = 54°, AB = 13.6
B. mA = 54°, mB = 36°, AB = 13.6
C. mA = 36°, mB = 54°, AB = 16.3
D. mA = 54°, mB = 36°, AB = 16.3
Example 5

66. Solve the right triangle
Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.
A. mA = 36°, mB = 54°, AB = 13.6
B. mA = 54°, mB = 36°, AB = 13.6
C. mA = 36°, mB = 54°, AB = 16.3
D. mA = 54°, mB = 36°, AB = 16.3
Example 5

67. End of the Lesson