# (15.) To find the distance from point A to point B across. a river, a base line AC is extablished. AC is 495 meters

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1 (15.) To find the distance from point A to point B across a river, a base line AC is extablished. AC is 495 meters long. Angles <BAC and <BCA are 89 o and 55 o respectively. Find the distance from A to B. Here is the diagram: A + B + C = B + 55 = 180 make substitutions B = 180 combine like terms subtract 144 from each side B = 36 subtract sin B sin C = b c sin 36 sin 55 = 495 c use the law of sines make substitutions

2 c sin 36 = 495 sin 55 cross multiply sin 36 sin 36 divide each side by sin 36 c = (495 sin 55)/(sin 36) c = 690 result: AB = 690 meters (16.) A ship at sea is sighted from two observation posts, A and B, on shore. Points A and B are 24 km apart. The measure of the angle <A between AB and the ship is 41 o. The angle at <B is 36 o. Find the distance from obser - vation post A to the ship. Here is the diagram:

3 A + B + C = C = 180 make substitutions C + 77 = 180 combine like terms subtract 77 from each side C = 103 subtract sin C sin B = c b use the law of sines sin 103 sin 36 = 24 b make substitutions b sin 103 = 24 sin 36 cross multiply sin 103 sin 103 divide each side by sin 103 b = (24 sin 36)/(sin 103) b = 14.5 (17.) Two points, M and N, are separated by a swamp. A baseline MK is established on one side of the swamp. MK is 180 m in length. The angles <NMK and <MKN are measured and found to be 44 o and 62 o, respectively. Find the distance between M and N. Here is the diagram:

4 M + N + K = N + 62 = 180 make substitutions N = 180 combine like terms subtract 106 from each side N = 74 subtract sin N sin K = n k use the law of sines sin 74 sin 62 = 180 k make substitutions k sin 74 = 180 sin 62 cross multiply sin 74 sin 74 divide each side by sin 74 k = (180 sin 62)/(sin 74) k = 165

5 result: MN = 165 (18.) Two angles of a triangle are 30 o and 55 o and the longest side is 34 m. Find the length of the shortest side. Here is the diagram: A + B + C = C = 180 make substitutions 85 + C = 180 combine like terms subtract 85 from each side C = 95 subtract sin C sin A =

6 c a use the law of sines sin 95 sin 30 = 34 a make the substitutions a sin 95 = 34 sin 30 cross multiply sin 95 sin 95 divide each side by sin 95 a = (34 sin 30)/(sin 95) a = 17 (19.) Two ranger stations located 10 km apart receive a distress call from a camper. Electronic equipment allows them to determine that the camper is at an angle of 71 o from the first station and 100 o from the second. Each of these angles has as one side the line segment connecting the stations. Which of the stations is closer to the camper? How far away is it from the camper? Here is the diagram:

7 A + B + C = C = 180 make substitutions C = 180 combine like terms subtract 171 from each side C = 9 subtract (i.) The 2nd station is closer to the camper than the 1st station. (ii.) sin C sin A = c a use the law of sines sin 9 sin 71 = 10 a make substitutions a sin 9 = 10 sin 71 cross multiply sin 9 sin 9 divide each side by sin 9 a = (10 sin 71)/(sin 9)

8 a = 60.4 (20.) A tree stands at point C across a river from point A. A baseline AB is established on one side of the river. The measure of AB is 80 m. The measure of <BAC is 54 o. and that of <CBA is 74 o. The angle of elevation of the top of the tree from A measures 10 o. Find the height of the tree. (i.) Here is the diagram of the triangle on the ground: A + B + C = C = 180 make substitutions C = 180 combine like terms

9 subtract 128 from each side C = 52 subtract sin B sin C = b c use the law of sines sin 74 sin 52 = b 80 make substitutions b sin 52 = 80 sin 74 cross multiply sin 52 sin 52 divide each side by sin 52 b = (80 sin 74)/(sin 52) b = 97.6 (ii.) Here is the diagram of the elevation of the tree: tan 10 = h/97.6 h/97.6 = tan 10 h = 97.6 tan 10 use this trig equation just rearrange multiply each side by 97.6,

10 h = 17.2 (21.) From the top and bottom of a tower 28 m high, the angles of depression of a ship are 18 o and 14 o, respectively. What is the distance of the ship from the foot of the tower? Here is the diagram: A + B + C = B + 72 = 180 make substitutions B = 180 combine like terms subtract 176 from each side B = 4 subtract

11 sin B sin C = b c use the law of sines sin 4 sin 72 = 28 c make substitutions c sin 4 = 28 sin 72 cross multiply sin 4 sin 4 divide each side by sin 4 c = (28 sin 72)/(sin 4) c = (21b.) The lengths of two sides and one diagonal of a parallelogram are 8 m, 13 m, and 20 m, respectively. What is the measure of each angle of the parallelogram? Here is the diagram:

12 AC 2 = AD 2 + DC 2-2(AD)(DC)cos D use the law of cosines 20 2 = (8)(13) cos D make substitutions 400 = cos D multiply and add subtract 233 from each side 167 = -208 cos D subtract divide each side by -208 cos D = -167/208 D = arccos (-167/208) and just rearrange like this take the arccos of each side D = = 36.6 subtract from 180 results: <D and <B are each 143.4; and <A and <C are each 36.6 (22.) The lengths of the sides of a triangle are 8, 9 and 13 cm. Determine whether the largest angle is acute or obtuse. c 2 = a 2 + b 2-2ab cos C use the law of cosines 13 2 = (8) 2 + (9) 2-2(8)(9)cos C make substitutions 169 = cos C multiply 169 = cos C combine like terms subtract 145 from each side 24 = -144 cos C subtract divide each side by -144

13 -1/6 = cos C reduce and result: <C is obtuse (24.) A triangular lot has sides of 215, 185, and 125 meters. Find the measures of the angles at its corners. c 2 = a 2 + b 2-2ab cos C use the law of cosines = (125)(185)cos C make substitutions = cos C multiply and add subtract from each side 12,000 = cos C subtract divide ea side by this cos C = /46250 C = arccos (-12000/46250) and rearrange like this take the arccos of each side C = 105 sin C sin B = c b use the law of sines sin 105 sin B = make substitutions 185 sin 105 = 215 sin B cross multiply divide each side by 215 sin B = (185 sin 105)/215

14 B = arcsin [(185 sin 105)/215] take the arcsin of ea side B = 56.2 A + B + C = 180 A = 180 A = 180 make substitutions combine like terms subtract from each side A = 18.8 subtract results: A = 18.8 ; B = 56.2 ; C = 105 (25.) From point C, both ends A and B of a railroad tunnel are visible. If AC = 165 m, BC = 115 m, and C = 74 o, find AB, the length of the tunnel. Here is the diagram:

15 c 2 = a 2 + b 2-2ab cos C use the law of sines c 2 = (115)(165)cos 74 make substitutions c 2 = c = 173 take square roots (26.) The distances from a boat B to two points A and C on the shore are known to be 100 m and 80 m respectively, and <ABC = 55 o. Find AC. b 2 = a 2 + c 2-2ac cos B use the law of cosines b 2 = (100)(80) cos 55 make substitutions b 2 = b = 85 take square roots (27.) The radius of a circle is 20 cm. Two radii, OX and OY, form an angle of 115 o. How long is the chord XY? c 2 = a 2 + b 2-2ab cos C use the law of cosines c 2 = (20)(20)cos 115 make substitutions c 2 = 1138 c = 33.7 take square roots (28.) Two sides and a diagonal of a parallelogram are 7, 9, and 15 feet respectively. Find the measures of the angles of the parallelogram. c 2 = a 2 + b 2-2ab cos C use the law of cosines

16 15 2 = (7)(9)cos C make substitutions 225 = cos C multiply and add -225 = cos C multiply thru by add 130 to each side 95 = 126 cos C add 126 cos C = 95 rearrange like this divide each side by 126 cos C = (95/126) C = arccos (95/126) C = 41 take arccos of each side = 139 subtract from 180 results: the angles of the parallelogram are 139, 41, 139, & 41 (53.) A salvage ship using sonar finds the angle of depression of wreckage on the ocean floor to be 13 o. The charts show that in this region the ocean floor is 35 m below the surface. How far must a diver lowered from the salvage ship travel along the ocean floor to reach the wreckage? tan 13 = 35/d use this trig equation d tan 13 = 35 multiply each side by d, tan 13 tan 13 divide each side by tan 13

17 d = and (56.) Two travelers, driving along a highway, spot their destination in the distance at point C. They stop at the side of the road to measure angle A. (45 o ) They then travel 5 more miles along the road, (AB), stop again, and measure angle B. (120 o ) How far are the travelers now from their destination? (find AC) A + B + C = C = 180 make substitutions C = 180 add subtract 165 from each side C = 15 subtract sin C sin B = c b use the law of sines sin 15 sin 120 = 5 b make the substitutions b sin 15 = 5 sin 120 cross multiply sin 15 sin 15 divide each side by sin 15 b = (5 sin 120)/(sin 15) b = 16.7

18 (23.) A kite string 90 m long makes a 48 o angle with the horizontal. Find, to the nearest meter, the distance from the kite to the ground. sin 48 = h/90 use this trig equation h/90 = sin 48 rearrange h = 90 sin 48 h = multiply each side by 90 and (24.) Two lighthouses at points A and B are 40 km apart. Each has visual contact with a freighter at point C. <A = 20 o and <B = 115 o. Find AC. A + B + C = C = 180 make substitutions C = 180 combine like terms subtract 135 from each side C = 45 subtract sin C sin B = c b use the law of sines

19 sin 45 sin 115 = 40 b make substitutions b sin 45 = 40 sin 115 cross multiply sin 45 sin 45 divide each side by sin 45 b = (4 sin 115)/(sin 45) b = 5

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