822 CHAPTER 8 Applications of Trigonometry The number r is the absolute value (or modulus) of x + yi, and u is the argument of x + yi. In this section, we choose the value of u in the interval 30°, 360°2. Any angle coterminal with u also could serve as the argument. Trigonometric (Polar) Form of a Complex Number The trigonometric form (or polar form) of the complex number x + yi is r 1cos U +i sin U2. The expression cos u + i sin u is sometimes abbreviated cis U. Using this notation, r 1cos U +i sin U2 is written r cis U. Converting from Rectangular toTrigonometric Form Step 1 Sketch a graph of the number x + yi in the complex plane. Step 2 Find r by using the equation r = 2x2 + y2. Step 3 Find u by using the equation tan u = y x , where x ≠0, choosing the quadrant indicated in Step 1. CAUTION Errors often occur in Step 3. Be sure to choose the correct quadrant for U by referring to the graph sketched in Step 1. EXAMPLE 2 Converting fromTrigonometric Form to Rectangular Form Write 21cos 300° + i sin 300°2 in rectangular form. ALGEBRAIC SOLUTION 21cos 300° + i sin 300°2 = 2¢ 1 2 - i 23 2 ≤ cos 300° = 1 2 ; sin 300° = - 23 2 = 1 - i23 Distributive property Note that the real part is positive and the imaginary part is negative. This is consistent with 300° being a quadrant IV angle. For a 300° angle, the reference angle is 60°. Thus the function values cos 300° and sin 300° correspond in absolute value to those of cos 60° and sin 60°, with the first of these equal to 1 2 and the second equal to - 23 2 . GRAPHING CALCULATOR SOLUTION In Figure 49, the first result confirms the algebraic solution, where an approximation for -23 is used for the imaginary part (from the second result). The TI-84 Plus also converts from polar to rectangular form, as seen in the third and fourth results. Figure 49 S Now Try Exercise 37. Converting between Rectangular and Trigonometric Forms To convert from rectangular form to trigonometric form, we use the following procedure.
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