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Review - Test 2 MATH 2412 - Spring, 2017 Morales, Instructor Use the information given about the angle , 0 2 , to find the exact value of the indicated trigonometric function. 3 Find cos (2 ). 12) sin = , 0 < < 5 2 Simplify the trigonometric expression by following the indicated direction. 6 cos2 + 7 cos + 1 1) Factor and simplify: cos2 - 1 Complete the identity. sin sin 2) 1 + sin 1 - sin 3) sec sin tan 4) cos =? -1=? - cos sin2 =? Find the exact value of the trigonometric function. 5) sin 15° (Hint: Use 15 = 45 - 30) 2 < < < 3 ; cos 2 15) sin =- 3 3 , < 5 2 =- < < 3 2 Find cos <2 Find sin 2 2 Solve the equation on the interval 0 3 20) sin (4 ) = 2 Find cos ( + ). < 3 , 5 19) cos (2 ) = Find the exact value under the given conditions. 21 3 , 0 < < ; cos = , 0 < < 10) sin = 29 2 5 2 3 , 4 =- Find sin (2 ). . 2 2 . Solve the equation. Give a general formula for all the solutions. 3 18) sin = 2 9) cos 20° cos 40° - sin 20° sin 40° = 14) cos <2 17) cos 22.5° 5 (Hint: = 75, and 75 = 30 + 45) 12 Find the exact value of the expression. 8) sin 25° cos 35° + cos 25° sin 35° 11) tan = Use the Half-angle Formulas to find the exact value of the trigonometric function. 16) sin 22.5° 6) sin 165° (Hint: Use 165 = 45 + 120) 5 7) cos 12 24 3 , < 25 2 13) cos 24 , 25 21) cos Find sin ( + ). -1=0 22) 2 cos +1=0 23) cos2 + 2 cos 24) sin2 + sin 25) sin (2 ) + sin 1 +1=0 =0 =0 <2 . Solve the equation for solutions in the interval [0°, 360°). 26) sin 2 = cos 27) sin 2 = - Find the missing parts of the triangle. 35) B = 108.2° a = 298 cm b = 1351 cm 1 2 36) A = 124° a = 1204 cm b = 1304 cm Let triangle ABC have standard labeling. Given the following combination of angles and sides, decide whether solving the triangle results in the ambiguous case. 28) A, c, and a 37) B = 23.0° b = 16.41 a = 21 29) a, b, and c 38) C = 124.7° a = 5.90 km b = 10.61 km 30) b, B, and c Assume triangle ABC has standard labeling. Determine if AAS, ASA, SSA, SAS, or SSS is given and decide if the law of sines or the law of cosines should be used to solve the triangle. 31) a, b, and A 39) a = 7.0 in. b = 13.6 in. c = 16.4 in. Provide an appropriate response. 40) Explain, in your own words, the situation called "the ambiguous case of the law of sines." 32) A, b, and c Solve the triangle. 33) 41) Explain why the law of sines cannot be used to solve a triangle if we are given the lengths of two sides and the measure of the included angle. 23 m Find the area of the triangle using one of the area formulas. 42) a = 19.7 cm b = 15.9 cm c = 16.6 cm 34) 43) b = 7.9 in. A = 29.33° C = 71.33° 9 Solve. 2 44) Two tracking stations are on the equator 117 miles apart. A weather balloon is located on a bearing of N 34°E from the western station and on a bearing of N 13°E from the eastern station. How far is the balloon from the western station? Solve the problem. 45) To find the distance between two small towns, an electronic distance measuring (EDM) instrument is placed on a hill from which both towns are visible. If the distance from the EDM to the towns is 4.8 miles and 3.7 miles and the angle between the two lines of sight is 63°, what is the distance between the towns? Round your answer to the nearest tenth of a mile. 46) A room in the shape of a triangle has sides of length 12.6 yd, 15.7 yd, and 15.8 yd. If carpeting costs $20.15 per sq. yd, padding costs $2.50 per sq. yd, and there is no charge for installation, how much, to the nearest dollar, will it cost to carpet the room? 3 Answer Key Testname: MATH 2412 TEST 2 REVIEW SPRING 2017 1) 6 cos + 1 cos - 1 24) 0, , 2) -2 tan2 3) 0 4) cos3 25) 0, 5) 2( 3 - 1) 4 6) 2( 3 - 1) 4 7) 2( 3 - 1) 4 9) 1 2 24 10) 145 11) 38) 39) 44 125 40) 7 12) 25 13) - 336 625 14) - 5 5 15) - 10 10 16) 1 2 2- 2 17) 1 2 2+ 2 18) = 19) = 20) 3 8 41) + 2k , +k , 2 4 , , 3 3 26) {30°, 150°} 27) {105°, 165°, 285°, 345°} 28) Yes 29) No 30) Yes 31) SSA; law of sines 32) SAS; law of cosines 33) C = 103°, a = 10.3 m, b = 18.3 m 34) a 7.3, B 7.8°, C 142.2° 35) A = 12.1°, C = 59.7°, c = 1227 cm 36) No solution 37) A1 = 30°, C1 = 127°, c1 = 33.54; 3 2 8) 3 2 42) = = 2 + 2k 3 7 +k 8 A2 = 150°, C2 = 7°, c2 = 5.12 c = 14.8 km, A = 19.1°, B = 36.2° A = 24.81°, B = 54.61°, C = 100.58° The ambiguous case of the law of sines occurs when we are given the lengths of two sides of an oblique triangle and the measure of the angle opposite one of them (a non-included angle). In this case, it is possible that 0, 1, or 2 triangles exist. The law of sines requires that we know either one side and two angles or two sides and the angle opposite one of them (a non-included angle). 127 cm2 43) 14.7 in.2 44) 318 miles 45) 4.5 miles 46) $2060 2 7 7 13 5 , , , , , 12 6 3 12 6 12 3 , , 19 12 21) 0 2 4 , 22) 3 3 23) 4