823 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients See Example 3(a). The TI-84 Plus converts from rectangular form to polar form. The value of u in the second result is an approximation for 5p 6 , as shown in the third result. (b) See Figure 51. Because -3i = 0 - 3i, we have x = 0 and y = -3. r = 202 + 1-322 = 20 + 9 = 29 = 3 Substitute. We cannot find u by using tan u = y x because x = 0. However, the graph shows that the least positive value for u is 270°. -3i = 31cos 270° + i sin 270°2, or 3 cis 270° Trigonometric form S Now Try Exercises 49 and 55. EXAMPLE 3 Converting from Rectangular toTrigonometric Form Write each complex number in trigonometric form. (a) -23 + i (Use radian measure.) (b) -3i (Use degree measure.) SOLUTION (a) We start by sketching the graph of -23 + i in the complex plane, as shown in Figure 50. Next, we use x = -23 and y = 1 to find r and u. r = 2x2 + y2 = 4A -23 B 2 + 12 = 23 + 1 = 24 = 2 tan u = y x = 1 -23 = - 12 3 # 23 23 = - 23 3 Rationalize the denominator. Because tan u = - 23 3 , the reference angle for u in radians is p 6 . From the graph, we see that u is in quadrant II, so u = p - p 6 = 5p 6 . -23 + i = 2 acos 5p 6 + i sin 5p 6 b, or 2 cis 5p 6 x 0 –2 1 y = 1 r = 2 u = –√3 + i x = – √3 5p 6 y Figure 50 y x 1 1 0 r = 3 0 – 3i u = 270° Figure 51 Compare to the result in Example 3(b). The angle -90° is coterminal with 270°. The calculator returns u values between -180° and 180°. EXAMPLE 4 Converting betweenTrigonometric and Rectangular Forms Using Calculator Approximations Write each complex number in its alternative form, using calculator approximations as necessary. (a) 61cos 125° + i sin 125°2 (b) 5 - 4i SOLUTION (a) Because 125° does not have a special angle as a reference angle, we cannot find exact values for cos 125° and sin 125°. 61cos 125° + i sin 125°2 ≈61-0.5735764364 + 0.8191520443i2 ≈ -3.4415 + 4.9149i Four decimal places Use a calculator set to degree mode.
RkJQdWJsaXNoZXIy NjM5ODQ=