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NAME DATE 4-1 PERIOD Practice Right Triangle Trigonometry Find the exact values of the six trigonometric functions of θ. √ 55 8 3 , cos θ = − , sin θ = − θ 8 3 2. 24 , sin θ = − 7 θ 8 3 √ 55 8 , tan θ = − , csc θ = − 55 3 √ 8 √ 55 55 sec θ = − , cot θ = − 55 3 25 7 cos θ = − , 25 25 24 tan θ = − , csc θ = − , 7 24 25 7 sec θ = − , cot θ = − 7 24 24 Find the value of x. Round to the nearest tenth, if necessary. 3. 35.3 4. 9.4 26° x 19.2 56° x 63.1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. On a college campus, the library is 80 yards due east of the dormitory and the recreation center is due north of the library. The college is constructing a sidewalk from the dormitory to the recreation center. The sidewalk will be at a 56° angle with the current sidewalk between the dormitory and the library. To the nearest yard, how long will the new sidewalk be? 143 yd 6. If cot A = 8, find the exact values of the remaining trigonometric functions for the acute angle A. √ √ 65 8 √ 65 65 1 sin A = − , cos A = − , tan A = − , sec A = − , csc A = √ 65 65 65 8 8 Find the measure of angle θ. Round to the nearest degree, if necessary. 7. 2 4 8. θ θ 3 7 42° 55° Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. 9. " 22° $ 14 a ≈ 5.2, b ≈ 13.0, B = 68° 10. $ 7 " # 2 c c ≈ 7.3, A ≈ 16°, B ≈ 74° # 11. SWIMMING The swimming pool at Perris Hill Plunge is 50 feet long and 25 feet wide. If the bottom of the pool is slanted so that the water depth is 3 feet at the shallow end and 15 feet at the deep end, what is the angle of elevation at the bottom of the pool? about 13.5° Chapter 4 7 Glencoe Precalculus Lesson 4-1 1. NAME DATE 4-1 PERIOD Word Problem Practice Right Triangle Trigonometry 1. MONUMENTS The Leaning Tower of Pisa in Italy is about 55.9 meters tall and is leaning so it is only about 55 meters above the ground. At what angle is the tower leaning? 5. OBSERVATION A person standing 100 feet from the bottom of a cliff notices a tower on top of the cliff. The angle of elevation to the top of the cliff is 30°, and the angle of elevation to the top of the tower is 58°. How tall is the tower? about 10.3° about 102.3 ft 2. SUBMARINES A submarine that is 250 meters below the surface of the ocean begins to ascend at an angle of 22° from vertical. How far will the submarine travel before it breaks the surface of the water? 58° about 269.6 m 30° 100 ft 3. PHYSICS Suppose you are traveling in a car when a beam of light passes from the air to the windshield. The measure of the angle of incidence θi is 55°, and the measure of the angle of refraction sin θ θr is 35.25°. Use Snell’s Law, −i = n, 6. GEOMETRY The apothem of a regular polygon is the measure of the line segment from the center of the polygon to the midpoint of one of its sides. A circle is circumscribed about a regular hexagon with an apothem of 4.8 centimeters. sin θr to find the index of refraction n of the windshield to the nearest thousandth. 4.8 cm 4. CONSTRUCTION A 30-foot ladder leaning against the side of a house makes a 70.1° angle with the ground. a. Find the radius of the circumscribed circle. about 5.5 cm b. What is the length of a side of the hexagon? about 5.5 cm c. What is the perimeter of the hexagon? about 33 cm 30 ft 7. SKATEBOARD Suppose you want to construct a ramp for skateboarding with a 19° incline and a height of 4 feet. 70.1° a. How far up the side of the house does the ladder reach? a. Draw a diagram to represent the situation. about 28.2 ft b. What is the horizontal distance between the bottom of the ladder and the house? 4 ft 19° about 10.2 ft b. Determine the length of the ramp. about 12.3 ft Chapter 4 8 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. about 1.419 NAME DATE 4-2 PERIOD Practice Degrees and Radians Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest thousandth. 1. 28.955° 2. -57.3278 28° 57' 18" 3. 32° 28' 10" 32.469° -57° 19' 40.08" 4. -73° 14' 35" -73.243° Write each degree measure in radians as a multiple of π and each radian measure in degrees. 5π − 5. 25° 3π 7. − 4 13π − 6. 130° 36 18 5π 8. − 3 135° 300° Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative angle coterminal with the given angle. 9. 43° 7π 10. − − 43° + 360n° y 4 y y 43° x 403° − + 2nπ − 7π 43° -317° 4 y π 4 x - 7π 4 - 15π 4 x x −, − 15π Sample answers: π − Sample answers: 403°, −317° 4 4 Find the length of the intercepted arc with the given central angle measure in a circle of the given radius. Round to the nearest tenth. 11. 30°, r = 8 yd 4.2 yd −, r = 10 in. 36.7 in. 12. 7π 6 Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation. 4 π rad/s 24 rev/min 13. ω = − 5 14. V = 32 m/s, 100 rev/min 3.06 m 15. On a game show, a contestant spins a wheel. The angular speed of the wheel π was ω = − radians per second. If the wheel maintained this rate, what would be 3 the rotation in revolutions per minute? 10 rev/min Find the area of each sector. π 16. θ = − , r = 14 in. 51.3 in2 6 14 in. Chapter 4 7π 17. θ = − , r = 4 m 44.0 m2 4 π 6 7π 4 12 4m Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. - 7π 4 NAME DATE 4-3 PERIOD Practice Trigonometric Functions on the Unit Circle The given point lies on the terminal side of an angle θ in standard position. Find the values of the six trigonometric functions of θ. 1. (−1, 5) 2. (7, 0) √ 26 5 √ 26 sin θ = − , cos θ = - − , 26 26 √ 26 tan θ = −5, csc θ = − , 3. (−3, −4) sin θ = - − , cos θ = - − , 5 1 sec θ = - √ 26 , cot θ = − − 5 4. (1, −2) tan θ = − , csc θ = - − , 3 √ 5 5 4 3 3 4 sec θ = - − , cot θ = − sec θ = 1, cot θ = undefined 5. (−3, 1) 2 √ 5 5 5 4 tan θ = 0, csc θ = undefined, 5 3 4 sin θ = 0, cos θ = 1, 6. (2, −4) √ 10 3 √ 10 10 1 10 2 √ 5 √5 sin θ = - − , cos θ = − , sin θ = − , cos θ = - − , sin θ = - − , cos θ = − , tan θ = -2, csc θ = - − , tan θ = - − , csc θ = √ 10 , tan θ = -2, csc θ = - − , sec θ = √ 5 , cot θ = - − sec θ = - − , cot θ = -3 , cot θ = - − sec θ = √5 5 √5 5 2 1 2 3 √ 10 3 5 5 √5 2 1 2 Sketch each angle. Then find its reference angle. y Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. θ = 330° θ = 30° π − 4 4 0 θ = π 6 6 θ = π π − π 12. − − 45° y 3 3 x 0 y θ = 135° θ = 45° 4 x x 0 θ = π x 0 6 4 y θ = 7π y θ = 7π θ = - 3π 4 11. 135° 4 6 x x 0 π − 7π 9. − 0 θ =-π 3 4 θ = π 3 Find the exact value of each expression. If undefined, write undefined. 13. csc 90° 1 14. tan 270° 3π 16. cos − 0 π 17. sec − − 2 undefined √ 2 ( 4) 15. sin (−90°) 5π 18. cot − 6 −1 - √3 19. PENDULUMS The angle made by the swing of a pendulum and its vertical resting position can be modeled by θ = 3 cos πt, where t is time measured in seconds and θ is measured in radians. What is the angle made by the pendulum after 6 seconds? 3 rad Chapter 4 17 Glencoe Precalculus Lesson 4-3 4 y 7π 10. − π − 3π 8. − − 30° 7. 330°