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PRACTICE TEST 3 Name___________________________________ YOU MUST SHOW ALL WORK TO RECIEVE FULL CREDIT SHOW YOUR WORK. Establish the identity. 1) cot · sec = csc 1) 2) (1 - cos x)(1 + cos x) = sin2 x 2) 3) tan u - 1 1 - cot u = tan u + 1 1 + cot u 3) 4) 1 + csc x = cos x + cot x sec x 4) 5) sin x sin x + = 2 csc x 1 - cos x 1 + cos x 5) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 11 6) sin 12 A) 7) sin 6 B) - 6+ 4 2 C) 2+ 4 6 D) 64 2 7) 12 A) 8) tan 24 6) 2( 3 - 1) B) 2( 3 - 1) 4 C) - 2( 3 - 1) 2( 3 - 1) 4 D) - 13 12 8) A) 2 + 3 B) -2 - 3 C) 3-2 D) 2 - 3 9) sin 15° A) 9) 2( 3 + 1) 4 10) tan 75° A) - 3 - 2 B) - 2( 3 + 1) 4 2( 3 - 1) 4 C) - D) 2( 3 - 1) 4 10) B) 3+2 C) 11) sin 185° cos 65° - cos 185° sin 65° 3 1 A) B) 2 2 3-2 D) - 3 + 2 11) 37 C) 12 1 3 D) 2 12) sin 15° cos 105° + cos 15° sin 105° 3 1 A) B) 2 2 12) 3 2 C) D) 1 4 13) cos 35° cos 25° - sin 35° sin 25° 3 A) 14) cos 2 2 14) B) -1 C) 0 D) 1 tan 5° + tan 25° 1 - tan 5° tan 25° A) 2 16) 1 D) 2 5 2 5 2 cos sin - sin 18 9 18 9 A) 15) 13) 3 C) 2 1 B) 4 15) 3 B) C) 1 2 3 3 D) tan 40° + tan 110° 1 - tan 40° tan 110° A) - 1 2 16) B) - 3 3 C) - 3 D) -2 1 3 17) sin cos-1 - sin-1 2 2 A) 2 3 2 18) cos tan-1 17) B) 0 C) 3 3 D) 1 4 3 - sin-1 3 5 A) 1 18) B) 2 3 5 C) 24 25 D) 2 6 5 2 1 19) sin sin-1 + cos-1 3 3 A) 2 + 2 10 9 20) tan tan-1 A) 19) B) 2 3 5 C) 2 3 + 2 10 9 D) 2 6 5 3 1 + sin-1 4 2 9+4 3 12 - 3 3 20) B) 2 3 5 C) 2 2 3 + 2 10 9 D) 2 6 5 Use the information given about the angle , 0 2 , to find the exact value of the indicated trigonometric function. 5 , 0< < Find cos(2 ). 21) sin = 21) 13 2 A) 22) cos 119 169 = A) - B) 3 3 , < 5 2 24 , 7 A) - 336 625 24) csc =- A) - 25) sec A) 26) sin A) 27) sin A) <2 B) - < < 3 2 3 < 2 D) 7 25 C) 336 625 D) - <2 7 25 24) C) 4 21 25 D) 17 25 25) C) - 17 25 D) 4 21 25 26) 7 25 C) - 24 25 D) 24 25 Find sin(2 ). B) - 27) 24 25 C) 24 25 D) - Find the exact value of the expression. 3 28) sin 2 cos-1 2 A) 3 527 625 Find cos(2 ). B) - 4 , 5 24 25 23) -4 21 25 <2 7 25 =- C) Find sin(2 ). B) 4 3 , < 5 2 24 25 -4 21 25 >0 17 25 =- 9 13 Find cos(2 ). B) 5 21 , csc 21 D) 22) 527 625 >0 17 25 =- 119 169 Find sin(2 ). B) 5 , tan 2 C) - Find sin(2 ). 7 25 = 23) tan 120 169 7 25 28) B) - 3 2 C) 3 3 2 D) 1 2 29) sin 2 sin-1 2 2 29) A) 0 B) 1 30) cos 2 sin-1 A) 3 C) D) 1 2 5 13 30) 119 169 B) 10 13 C) - 12 13 D) 2 5 + 10 13 12 31) cos 2 tan-1 5 31) 3 13 A) - B) - 119 169 C) 10 13 D) - 144 169 Use the information given about the angle , 0 2 , to find the exact value of the indicated trigonometric function. 1 Find sin . 32) sin = , 0 < < 32) 4 2 2 8 + 2 15 4 A) 33) sin 1 , tan 4 = A) 35) cos A) >0 8 + 2 15 4 A) 34) csc B) =- >0 5 6 = 6 4 < B) 8+2 4 10 4 D) 6 4 34) 18 - 6 5 6 3+ 6 D) - 5 . 35) 15 C) 8-2 4 15 Use the Half-angle Formulas to find the exact value of the trigonometric function. 36) sin 22.5° 1 1 1 2+ 2 2+ 2 2- 2 A) B) C) 2 2 2 37) tan 165° A) -2 + 10 4 . C) - 2 D) 33) C) 2 8 - 2 15 4 . 18 + 6 5 6 Find cos 2 2 8 - 2 15 4 Find cos B) 1 , 0< 4 C) Find cos B) 3 , tan 2 6 4 D) 10 4 1 D) 2 36) 2- 2 37) 3 B) -2 - 3 C) 2 + 4 3 D) 2 - 3 38) tan 75° A) -2 + 39) sin 3 B) -2 - 3 C) 2 + 3 D) 2 - 3 39) 12 A) 40) tan 38) 1 2 1- 3 B) 1 2 1- 3 C) 1 2 2+ 3 D) 1 2 2- 40) 8 A) 1 + 2 B) -1 - 2 C) -1 + 2 D) 1 - 2 Express the product as a sum containing only sines or cosines. 41) sin(4 ) cos(2 ) 1 1 A) [cos(6 ) - cos(2 )] B) [sin(6 ) + cos(2 )] 2 2 C) sin cos(8 2 ) D) 42) sin(7 ) sin(4 ) 1 [cos(11 ) - cos(3 )] 2 D) 43) cos(6 ) cos(5 ) 1 A) [cos(11 ) - cos ] 2 C) 1 [ cos 2 1 [sin(6 ) + sin(2 )] 2 42) 1 [sin(11 ) + cos(3 )] 2 43) B) cos2 (30 2 ) + cos(11 )] D) 44) sin(3 ) sin(7 ) 1 [cos(11 ) - sin ] 2 1 B) [- cos(4 ) - cos(10 )] 2 A) sin2 (21 2 ) C) 41) 1 B) [ cos(3 ) - cos(11 )] 2 A) sin2 (28 2 ) C) 3 1 [cos(4 ) - cos(10 )] 2 D) 1 [cos(10 ) - sin(4 )] 2 Express the sum or difference as a product of sines and/or cosines. 45) sin(7 ) + sin(3 ) A) 2 cos(5 ) sin(2 ) B) 2 sin(5 ) cos(2 ) C) 2 sin(5 ) sin(2 ) D) 2 sin(10 ) 46) cos(5 ) + cos(3 ) A) 2 sin(4 ) sin B) 2 cos(4 ) cos C) 2 cos(4 ) sin 5 44) 45) D) 2 cos(4 ) 46) 47) sin(4 ) - sin(6 ) A) 2 cos(4 ) cos(5 ) C) 2 sin(5 ) cos 48) cos B) -2 sin D) -2 sin 47) cos(5 ) 9 7 + cos 2 2 A) 2 cos(4 ) 49) sin(4 ) - sin(2 ) A) sin cos(3 ) 48) B) 2 cos(4 ) cos 2 B) 2 sin(3 ) cos C) 2 sin(4 ) sin D) 2 sin(4 ) sin C) sin(3 ) cos D) 2 sin 2 cos(3 ) Solve the problem. 50) On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by y = sin (2 lt) and y = sin (2 ht) where l and h are the low and high frequencies (cycles per second) shown on the illustration. 49) 50) The sound produced is thus given by y = sin (2 lt) + sin (2 ht) Write the sound emitted by touching the 6 key as a product of sines and cosines. A) y = 2 sin(439 t) cos(1979 t) B) y = 2 sin(2247 t) cos(707 t) C) y = 2 sin(707 t) cos(2247 t) D) y = 2 sin(1979 t) cos(439 t) 51) A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the lake 90 feet from a piling that is directly across from a pier on the other side of the lake. From his transit, the angle between the piling and the pier is 40°. What is the distance between the piling and the pier to the nearest foot? A) 58 ft B) 107 ft C) 76 ft D) 69 ft 51) 52) A radio transmission tower is 240 feet tall. How long should a guy wire be if it is to be attached 15 feet from the top and is to make an angle of 20° with the ground? Give your answer to the nearest tenth of a foot. A) 239.4 ft B) 657.9 ft C) 701.7 ft D) 255.4 ft 52) 53) A building 260 feet tall casts a 100 foot long shadow. If a person looks down from the top of the building, what is the measure of the angle between the end of the shadow and the vertical side of the building (to the nearest degree)? (Assume the person's eyes are level with the top of the building.) A) 21° B) 69° C) 23° D) 67° 53) 6 54) John (whose line of sight is 6 ft above horizontal) is trying to estimate the height of a tall oak tree. He first measures the angle of elevation from where he is standing as 35°. He walks 30 feet closer to the tree and finds that the angle of elevation has increased by 12°. Estimate the height of the tree rounded to the nearest whole number. A) 90 ft B) 61 ft C) 86 ft D) 67 ft Solve the triangle. 55) 54) 55) 25° 125° 4 A) C = 30°, a = 4.73, c = 7.75 C) C = 30°, a = 7.75, c = 4.73 B) C = 25°, a = 4.73, c = 7.75 D) C = 35°, a = 7.75, c = 4.73 56) A = 20°, B = 10°, a = 1 A) C = 150°, b = 1.46, c = 0.51 C) C = 150°, b = 1.46, c = -0.49 B) C = 150°, b = 0.51, c = 1.46 D) C = 150°, b = 1.51, c = 1.46 56) Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. 57) a = 7, b = 9, B = 49° 57) A) one triangle B) two triangles A = 35.94°, C = 95.06°, c = 11.88 A1 = 76.01°, C1 = 54.99°, c1 = 7.60 or A2 = 103.99°, C2 = 27.01, c2 = 12.14 D) no triangle C) one triangle A = 76.01°, C = 54.99°, c = 7.60 58) b = 3, c = 7, B = 80° A) one triangle C = 41°, A = 59°, a = 14 C) one triangle B = 40°, A = 60°, a = 10 B) one triangle C = 39°, A = 61°, a = 12 D) no triangle 59) a = 10, b = 7, B = 10° A) one triangle A = 165.64°, C = 4.36°, c = 3.06 B) two triangles A1 = 14.36°, C1 = 155.64°, c1 = 16.63 or 58) 59) A2 = 165.64°, C2 = 4.36°, c2 = 3.06 D) no triangle C) one triangle A = 14.36°, C = 155.64°, c = 16.63 60) A = 30°, a = 14, b = 28 A) B = 60°, C = 60°, c = 24.2 C) B = 90°, C = 60°, c = 24.2 B) B = 60°, C = 90°, c = 24.2 D) no triangle 7 60) 61) A = 65°, a = 5, b = 7 A) one triangle B = 32°, C = 83°, c = 14 C) one triangle A = 33°, C = 82°, c = 12 B) one triangle B = 34°, C = 81°, c = 16 D) no triangle 62) C = 35°, a = 18.7, c = 16.1 A) two triangles A1 = 42°, B1 = 103°, b1 = 27.4; B) one triangle A = 42°, B = 103°, b = 27.4 A2 = 138°, B2 = 7°, b2 = 3.4 C) two triangles A1 = 103°, B1 = 42°, b1 = 27.4; 61) 62) D) no triangle A2 = 7°, B2 = 138°, b2 = 3.4 Solve the problem. 63) A ship at sea, the Admiral, spots two other ships, the Barstow and the Cauldrew and measures the angle between them at be 45°. They radio the Barstow and by comparing known landmarks, the distance between the the Admiral and the Barstow is found to be 323 meters. The Barstow reports an angle of 59° between the Admiral and the Cauldrew. To the nearest meter, what is the distance between the Barstow and the Cauldrew? A) 49 m B) 81 m C) 235 m D) 266 m 64) It is 4.7 km from Lighthouse A to Port B. The bearing of the port from the lighthouse is N73°E. A ship has sailed due west from the port and its bearing from the lighthouse is N31°E. How far has the ship sailed from the port? Round your answer to the nearest 0.1 km. A) 2.7 km B) 3.1 km C) 3.5 km D) 3.7 km Solve the triangle. 65) 63) 64) 65) 3 110° 4 A) c = 5.76, A = 29.3°, B = 40.7° C) c = 4.76, A =40.7°, B = 29.3° B) c = 6.76, A = 29.3°, B = 40.7° D) c = 5.76, A = 40.7°, B = 29.3° 66) b = 4, c = 5, A = 95° A) a = 6.67, B = 48.3°, C = 36.7° C) a = 7.67, B = 36.7°, C = 48.3° B) a = 5.67, B = 48.3°, C = 36.7° D) a = 6.67, B = 36.7°, C = 48.3° 8 66) 67) 67) 9 6 4 A) A = 127.2°, B = 32.1°, C = 20.7° C) A = 127.2°, B = 20.7°, C = 32.1° B) A = 32.1°, B = 20.7°, C = 127.2° D) A = 32.1°, B = 127.2°, C = 20.7° 68) a = 5, b = 5, c = 3 A) A = 72.5°, B = 35°, C = 72.5° C) A = 35°, B = 72.5°, C = 72.5° B) A = 73.5°, B = 73.5°, C = 33° D) A = 72.5°, B = 72.5°, C = 35° Solve the problem. 69) In flying the 81 miles from Champaign to Peoria, a student pilot sets a heading that is 8° off course and maintains an average speed of 88 miles per hour. After 15 minutes, the instructor notices the course error and tells the student to correct his heading. Through what angle will the plane move to correct the heading and how many miles away is Peoria when the plane turns? A) 169°; 72.67 mi B) 11°; 59.29 mi C) 11°; 72.67 mi D) 169°; 59.29 mi 68) 69) 70) Two points A and B are on opposite sides of a building. A surveyor selects a third point C to place a transit. Point C is 50 feet from point A and 61 feet from point B. The angle ACB is 57°. How far apart are points A and B? A) 88.8 ft B) 67.5 ft C) 97.7 ft D) 53.8 ft 70) 71) The distance from home plate to dead center field in a certain baseball stadium is 410 feet. A baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it from first base to dead center field? A) 477.9 ft B) 335.1 ft C) 352.2 ft D) 387.4 ft 71) 9 Answer Key Testname: PRACTICETEST3 1) cot · sec = cos sin · 1 cos = 1 sin = csc 2) (1 - cos x)(1 + cos x) = 1 - cos2 x = sin2 x 1 1 - cot u -1 cot u cot u tan u - 1 1 - cot u 3) = = = tan u + 1 1 1 + cot u 1 + cot u +1 cot u cot u 4) 1 + csc x 1 cos x (sin x + 1) cos x sin x cos x = cos x 1 + = = + = cos x + cot x. sec x sin x sinx sin x sin x 5) sin x sin x sin x[(1 + cos x)+(1 - cos x)] 2 sin x 2 sin x + = = = = 2 csc x. 1 - cos x 1 + cos x (1 - cos x)(1 + cos x) 2 1 - cos x sin2 x 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) A B D D B D C D C D B B C A A A B C D B B B C B A B C A C D C A C D D D B 10 Answer Key Testname: PRACTICETEST3 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66) 67) 68) 69) 70) 71) C C B B B B D B C B A D C B A D B C D A C D A D A D B D C 11