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Five-Minute Check (over Lesson 8–3) 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 Over Lesson 8–3 Find x and y. A. B. 0% A B C D 0% D 0% B 0% A D. C C. A. B. C. D. Over Lesson 8–3 Find x and y. A. x = 5, y = 5 B. x = 5, y = 45 A. B. C. D. 0% C 0% B 0% A D. 0% D C. A B C D Over Lesson 8–3 The length of the diagonal of a square is centimeters. Find the perimeter of the square. A. 15 cm 0% B D. 60 cm A 0% A B C 0% D D C. 45 cm A. B. C. 0% D. C B. 30 cm Over Lesson 8–3 The side of an equilateral triangle measures 21 inches. Find the length of an altitude of the triangle. in. C. 14 in. in. 0% B D. A 0% A B C 0% D D B. 12 in. A. B. C. 0% D. C A. Over Lesson 8–3 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) 0% B D. (6, 2) A 0% A B C 0% D D C. (5, 3) C B. (–3, 3) A. B. C. 0% D. Over Lesson 8–3 The hypotenuse of a 30°-60°-90° triangle measures inches. What is the length of the side opposite the 30° angle? A. 10 in. D. 0% B A 0% A B C 0% D D C. C B. 20 in. A. B. C. 0% D. 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. • trigonometry • trigonometric ratio • sine • cosine • tangent • inverse sine • inverse cosine • inverse tangent Find Sine, Cosine, and Tangent Ratios A. Express sin L as a fraction and as a decimal to the nearest hundredth. Answer: Find Sine, Cosine, and Tangent Ratios B. Express cos L as a fraction and as a decimal to the nearest hundredth. Answer: Find Sine, Cosine, and Tangent Ratios C. Express tan L as a fraction and as a decimal to the nearest hundredth. Answer: Find Sine, Cosine, and Tangent Ratios D. Express sin N as a fraction and as a decimal to the nearest hundredth. Answer: Find Sine, Cosine, and Tangent Ratios E. Express cos N as a fraction and as a decimal to the nearest hundredth. Answer: Find Sine, Cosine, and Tangent Ratios F. Express tan N as a fraction and as a decimal to the nearest hundredth. Answer: A. Find sin A. A. B. 0% B A 0% A B C 0% D D D. C C. A. B. C. 0% D. B. Find cos A. A. B. 0% B A 0% A B C 0% D D D. C C. A. B. C. 0% D. C. Find tan A. A. B. 0% B A 0% A B C 0% D D D. C C. A. B. C. 0% D. D. Find sin B. A. B. 0% B A 0% A B C 0% D D D. C C. A. B. C. 0% D. E. Find cos B. A. B. 0% B A 0% A B C 0% D D D. C C. A. B. C. 0% D. F. Find tan B. A. B. 0% B A 0% A B C 0% D D D. C C. A. B. C. 0% D. 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. Use Special Right Triangles to Find Trigonometric Ratios Definition of cosine ratio Substitution Simplify. Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. 0% B A D. 0% A B C 0% D D C. A. B. C. 0% D. C B. 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. Estimate Measures Using Trigonometry Multiply each side by 60. Use a calculator to find y. KEYSTROKES: 60 SIN 7 ENTER 7.312160604 Answer: The treadmill is about 7.3 inches high. 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. B. C. D. C. 16 in. 0% B 0% A D. 15 in. 0% C B. 11 in. A B C D 0% D A. 1 in. Find Angle Measures Using Inverse Trigonometric Ratios Use a calculator to find the measure of P to the nearest tenth. 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: 2nd [COS] ( 13 ÷ 19 ) ENTER 46.82644889 Answer: So, the measure of P is approximately 46.8°. Use a calculator to find the measure of D to the nearest tenth. A. 44.1° 0% B 0% A D. 57.2° A B C 0% D D C. 55.4° A. B. C. 0% D. C B. 48.3° Solve a Right Triangle Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. Solve a Right Triangle Step 1 Find mA by using a tangent ratio. Definition of inverse tangent 29.7448813 ≈ mA Use a calculator. So, the measure of A is about 30. Solve a Right Triangle Step 2 Find mB using complementary angles. mA + mB = 90 30 + mB ≈ 90 mB ≈ 60 Definition of complementary angles mA ≈ 30 Subtract 30 from each side. So, the measure of B is about 60. 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 ≈ AB Use a calculator. Solve a Right Triangle So, the measure of AB is about 8.06. Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06 Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. A. mA = 36°, mB = 54°, AB = 13.6 0% B A D. mA = 54°, mB = 36°, AB = 16.3 0% 0% 0% D C. mA = 36°, mB = 54°, AB = 16.3 A. B. C. D. C B. mA = 54°, mB = 36°, AB = 13.6 A B C D