7.1 21. C = 65%, a = 25, b = 12 Using the Law of Sines In Exercises 1-10, use the Law 22. B = 48, a = 18, c = 12 of Sines to solve the triangle. If two solutions exist, find both. 23. Geometry The lengths of the diagonals of a parallelogram are 10 feet and 16 feet. Find the lengths 1. A = 320, B = 50%, a = 16 of the sides of the parallelogram when the diagonals 2. A = 38, B = 58, a = 12 intersect at an angle of 28. 3. B = 16, C = 98, a = 8.4 24. Surveying To approximate the length of a marsh, a 4. B = 95, C = 450, a = 104.8 surveyor walks 425 meters from point A to point B. Then the surveyor turns 65 and walks 300 meters to point C 5. A = 60 15', B = 45 30', b = 4.8 (see figure). Approximate the length AC of the marsh. 6. A = 82 45', B = 28 45', b = 40.2 7. A = 750, a = 51.2, b = 33.7 8. A = 150, a = 5, b = 10 425 m 9. B = 250, a = 6.2, b = 4 300 m 10. B = 150, a = 10, b = 3 Finding the Area of an Oblique Triangle In Exercises 11-14, find the area of the triangle having the indicated angle and sides. Using Heron's Area Formula In Exercises 25-28, use 11. A = 330, b = 7, c = 10 Heron's Area Formula to find the area of the triangle. 12. B = 80%, a = 4, c = 8 25. a = 3, b = 6, 13. C = 122, b = 18, a = 29 26. a = 15, b = 8, c = 10 14. C = 100%, a = 120, b = 74 27. a = 64.8, b = 49.2, c = 24.1 15. Height A tree stands on a hillside of slope 280 from 28. a = 8.55, b = 5.14, c = 12.73 the horizontal. From a point 75 feet down the hill, 7.3 the angle of elevation to the top of the tree is 45 (see Showing That Two Vectors Are Equivalent In Exercises figure). Find the height of the tree. 29 and 30, show that u and v are equivalent. 29. 30. (-3, 2) 4 (1, 4) -28 (0, -2) (-1, - 16. Surveying A surveyor finds that a tree on the opposite Finding the Component Form of a Vector In Exercises bank of a river has a bearing of N 22 30' E from a 31-34, find the component form and the magnitude of certain point and a bearing of N 150 W from a point the vector v. 400 feet downstream. Find the width of the river. 31. 32. 7.2 -5, 4) 6, 5) Using the Law of Cosines In Exercises 17-22, use the Law of Cosines to solve the triangle. (0, 1) 17. a = 9, b = 12, c = 20 -2 2 18. a = 7, b = 15, c = 19 + (2, -1) 19. A = 150, b = 10, c = 20 33. Initial point: (0, 10); terminal point: (7, 3) 20. a = 6.2, b = 6.4, c = 2.1 34. Initial point: (1, 5); terminal point: (15, 9)Vector Operations In Exercises 3540, find (a) u + v, (b) u v, (c) 3u, and (d) 2v + 5n. Then sketch each resultant vector. 35. u=(l,3),v=(3,6> 36. u= v= (4.714) 63.u=6i*j 64.u=8i*7j v=2i+5j v=3i4j Using Properties of Dot Products In Exercises 6568, use the vectors u = (6, 4) and v = (2, 1) to find the indicated quantity. State Whether the result is a vector or a scalar. 65. u - u 67. (u - vlv 66. \"v\" 3 68. (u - v)u Finding the Angle Between Two Vectors In Exercises 6972, find the angle 6 between the vectors. 69. u = (2J5, 74), v = (a/i, 1) 70. u = (3, J3), v = (4,345} 77r 7r: ' = . _. + . _. 71 u cos 4 1 sm 4 J _ 57:, + . 5n , v cos 6 1 sm 6 72. u = cos 45i + sin 45j v = cos 300i + sin 300j Finding the Angle Between Two Vectors In Exercises 7376, graph the vectors and nd the degree measure of the angle between the vectors. 73.u=4i+j 74.u=6i+2j v=i4j v=3ij 75. u : 7.2i 5.6j 76. u : 5.3i + 2.8j v =10i+ 3.1j v = 78.1i , 4j A Relationship of Two Vectors In Exercises 7780, determine whether 11 and v are orthogonal, parallel, or neither. 77. u=(12,28) 78. H2 (8,24) v 2 (6,9) v 2 (5,10) 79. u : (8,5) 80. 11 : (15,:31} v = (22,4) v = (20,268) Finding an Unknown Vector Component ln Exercises 8184, find the value of k such that the vectors u and v are orthogonal. 81.n2ikj 82.n22i+4j V2i+2j V26ikj 83.n2kij 84.n2ki2j v=2i2j v=i+4j Decomposing a Vector into Components In Exercises 8588, find the projection of 11 onto v. Then write 11 as the sum of two orthogonal vectors, one of which is projvu. 85.11 = (4.3), v = (8, 2> 86. u 2 (5, 6), v 2 (10,0) 87.11 = (27), v = (1, 71> 88. u 2 (*3,5), v 2 (*5,2) 89. Work Determine the work done by a crane lifting an 18,000-pound truck 48 inches. 90. Physics A 500-pound motorcycle is stopped on a hill inclined at 12. What force is required to keep the motorcycle from rolling back down the hill? Finding the Absolute Value of a Complex Number In Exercises 9194, plot the complex number and find its absolute value. 92. 10 41' 94. t 91. 5 + 31' 93. 7i Trigonometric Form ofa Complex Number In Exercises 9598, write the complex number in trigonometric form without using a calculator. 95. 4t 97. J: 96. 9 98. 3\\/3 + 31' Multiplying or Dividing Complex Numbers In Exercises 99102, perform the operation and leave the result in trigonometric form. 5 7t 71 21 7? .l _+'_' .+'.' 99 [2 cos 2 mm 2)][4(cos 4 1s1n4)_| 2n . . 2?? 2r . . 7r 100. [2(cos 3 + 18111 3 )][3(cos6 + 151116)] 20(cos 320 + i sin 320) 5(cos 80 + i sin 80) 102 3(cos 230 + isin 230) ' 9(cos 95 + isin 95) 101. Operations with Complex Numbers In Exercises 103106, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms and check your result with that of part (b). 103. (2 20(3 + 31') 104. (4 + 4i)(1 s) 3 3i 1 i 2 + 21 2 2: 105. 106. Finding a Power In Exercises 107110, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. 71 it 4 . + ' ' 107 [5(cos 12 ism 12)] 4H 4H 5 . + 108 [2(005 15 Min 15)] 109. (2 + 306 110. (1 08 Finding Square Roots In Exercises 111114, nd the square roots of the complex number. 111. J3 +1' 112. J? i 113. 41' 114. 9i Finding the nth Roots of a Complex Number In Exercises 115118, (a) use the theorem on page 590 to find the indicated roots of the complex number, (1)) represent each of the roots graphically, and (c) write each of the roots in standard form. 116. Fourth roots of 2561' 118. Fifth roots of 1024 115. Sixth roots of 729i 117. Cube roots of 8 Solving an Equation ln Exercises 119122, use the theorem on page 590 to find all solutions of the equation, and represent the solutions graphically. 119. x4 + 256 2 0 120. x5 2 32E 2 0 121. x3 + 81'2 0 122. x4 + 81 2 0 Conclusions True or False? In Exercises 123 and 124, determine whether the statement is true or false. Justify your answer. 123. The Law of Sines is true if one of the angles in the triangle is a right angle. 124. When the Law of Sines is used, the solution is always unique