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Trigonometric Ratios and Special Right Triangles

Learn how to find sine, cosine, and tangent ratios using special right triangles. Also, discover how to use inverse trigonometric ratios to find angle measures.

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Trigonometric Ratios and Special Right Triangles

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  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. The length of the diagonal of a square is centimeters. Find the perimeter of the square. (1-2) Find x and y. The side of an equilateral triangle measures 21 inches. Find the length of an altitude of the triangle. 5-Minute Check 1

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

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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

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

  13. Concept

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

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

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

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

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

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

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

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

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

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

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

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

  26. 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

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

  28. 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

  29. 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

  30. 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

  31. 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

  32. Concept

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

  34. ÷ ) 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

  35. 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

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

  37. 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

  38. 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

  39. 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

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

  41. 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

  42. End of the Lesson

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