821 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients ■ The Complex Plane and Vector Representation ■ Trigonometric (Polar) Form ■ Converting between Rectangular and Trigonometric Forms ■ An Application of Complex Numbers to Fractals ■ Products of Complex Numbers in Trigonometric Form ■ Quotients of Complex Numbers in Trigonometric Form The Complex Plane and Vector Representation Unlike real numbers, complex numbers cannot be ordered. One way to organize and illustrate them is by using a graph in a rectangular coordinate system. To graph a complex number such as 2 - 3i, we modify the coordinate system by calling the horizontal axis the real axis and the vertical axis the imaginary axis. Then complex numbers can be graphed in this complex plane, as shown in Figure 45. Each complex number a +bi determines a unique position vector with initial point 10, 02 and terminal point 1a, b2. y x 0 P 2 – 3i –3 2 Real axis Imaginary axis Figure 45 NOTE This geometric representation is the reason that a + bi is called the rectangular form of a complex number. (Rectangular form is also known as standard form.) Recall that 14 + i2 + 11 + 3i2 = 5 + 4i. Graphically, the sum of two complex numbers is represented by the vector that is the resultant of the vectors corresponding to the two numbers. See Figure 46. y x 0 1 + 3i 1 2 3 4 5 1 2 3 4 5 5 + 4i 4 + i Figure 46 EXAMPLE 1 Expressing the Sum of Complex Numbers Graphically Find the sum of 6 - 2i and -4 - 3i. Graph both complex numbers and their resultant. SOLUTION The sum is found by adding the two numbers. 16 - 2i2 + 1-4 - 3i2 = 2 - 5i Add real parts, and add imaginary parts. The graphs are shown in Figure 47. S Now Try Exercise 21. y x 0 2 4 6 –4 –2 –2 –6 –4 – 3i 2 – 5i 6 – 2i –4 Figure 47 y x y x r x + yi O P u Figure 48 Trigonometric (Polar) Form Figure 48 shows the complex number x + yi that corresponds to a vector OP with direction angle u and magnitude r. The following relationships among x, y, r, and u can be verified from Figure 48. Relationships among x, y, r, and U x =r cos U y =r sin U r =!x2 +y2 tan U = y x , if x ≠0 Substituting x = r cos u and y = r sin u into x + yi gives the following. x + yi = r cos u + 1r sin u2i Substitute. = r 1cos u + i sin u2 Factor out r.
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