831 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 106. The complex conjugate of r 1cos u + i sin u2 is r 1cos u - i sin u2. Use these trigonometric forms to show that the product of complex conjugates is always a real number. (Modeling) Electric Current Solve each problem. 107. The alternating current in an electric inductor is I = E Z amperes, where E is voltage and Z = R + XLi is impedance. If E = 81cos 20° + i sin 20°2, R = 6, and XL = 3, find the current. Give the answer in rectangular form, with real and imaginary parts to the nearest hundredth. 108. The current I in a circuit with voltage E, resistance R, capacitive reactance Xc, and inductive reactance XL is I = E R + 1XL - Xc2i . Find I if E = 121cos 25° + i sin 25°2, R = 3, XL = 4, and Xc = 6. Give the answer in rectangular form, with real and imaginary parts to the nearest tenth. (Modeling) Impedance In the parallel electric circuit shown in the figure, the impedance Z can be calculated using the equation Z = 1 1 Z1 + 1 Z2 , where Z1 and Z2 are the impedances for the branches of the circuit. 109. If Z1 = 50 + 25i and Z2 = 60 + 20i, approximate Z to the nearest hundredth. 110. Determine the angle u, to the nearest hundredth, for the value of Z found in Exercise 109. Relating Concepts For individual or collaborative investigation (Exercises 111 – 118) Consider the following complex numbers, and work Exercises 111 – 118 in order. w = -1 + i and z = -1 - i 111. Multiply w and z using their rectangular forms and the FOIL method. Leave the product in rectangular form. 112. Find the trigonometric forms of w and z. 113. Multiply w and z using their trigonometric forms and the method described in this section. 114. Use the result of Exercise 113 to find the rectangular form of wz. How does this compare to the result in Exercise 111? 115. Find the quotient w z using their rectangular forms and multiplying both the numerator and the denominator by the conjugate of the denominator. Leave the quotient in rectangular form. 116. Use the trigonometric forms of w and z, found in Exercise 112, to divide w by z using the method described in this section. 117. Use the result in Exercise 116 to find the rectangular form of w z . 118. How does the result in Exercise 117 compare to the result in Exercise 115? 50 V 25 V 20 V 60 V
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