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Learn about trigonometric ratios, special triangles, and solving real-world problems involving right triangles. Explore inverse trigonometric ratios and angle measurements. Includes practical exercises and step-by-step examples.

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 8–3) NGSSS 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. A B C D A. B. C. D. Find x and y. 5-Minute Check 1

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

  5. A B C D 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

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

  7. A B C D 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

  8. A B C D 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

  9. MA.912.T.2.1Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. MA.912.G.5.4 Solve real-world problems involving right triangles. NGSSS

  10. You used the Pythagorean Theorem to find missing lengths in right triangles. (Lesson 8–2) • Find trigonometric ratios using right triangles. • Use trigonometric ratios to find angle measures in right triangles. Then/Now

  11. trigonometry • trigonometric ratio • sine • cosine • tangent • inverse sine • inverse cosine • inverse tangent Vocabulary

  12. Concept

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

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

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

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

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

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

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

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

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

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

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

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

  25. 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

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

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

  28. 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

  29. KEYSTROKES: 607 7.312160604 SIN ENTER Estimate Measures Using Trigonometry Multiply each side by 60. Use a calculator to find y. Answer: The treadmill is about 7.3 inches high. Example 3

  30. A B C D 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

  31. Concept

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

  33. ÷ ) ENTER 2nd ( KEYSTROKES: [COS] 13 19 46.82644889 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. Answer: So, the measure of P is approximately 46.8°. Example 4

  34. A B C D Use a calculator to find the measure of D to the nearest tenth. A. 44.1° B. 48.3° C. 55.4° D. 57.2° Example 4

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

  36. Solve a Right Triangle Step 1 Find mA by using a tangent ratio. Definition of inverse tangent 29.7448813 ≈ mA Use a calculator. So, the measure of A is about 30. Example 5

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

  38. Solve a Right Triangle Step 3 Find AB by using the Pythagorean Theorem. (AC)2 + (BC)2 = (AB)2 Pythagorean Theorem 72 + 42 = (AB)2 Substitution 65= (AB)2 Simplify. Take the positive square root of each side. 8.06 ≈ABUse a calculator. Example 5

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

  40. A B C D 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

  41. End of the Lesson

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