825 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients (b) For z = 1 + 1i, we have the following. z2 - 1 = 1 + i22 - 1 Substitute for z; 1 + 1i = 1 + i = 11 + 2i + i22 - 1 = -1 + 2i i2 = -1 The absolute value is 21-122 + 22 = 25. Because 25 is greater than 2, the number 1 + 1i is not in the Julia set, and 11, 12 is not part of the graph. Square the binomial; 1x + y22 = x2 + 2xy + y2 S Now Try Exercise 71. Products of Complex Numbers in Trigonometric Form Using the FOIL method to multiply complex numbers in rectangular form, we find the product of 1 + i23 and -223 + 2i as follows. A1 + i23 B A -223 + 2iB = -223 + 2i - 2i132 + 2i223 FOIL method = -223 + 2i - 6i - 223 i2 = -1 = -423 - 4i Combine like terms. We can also find this same product by first converting the complex numbers 1 + i23 and -223 + 2i to trigonometric form using the method explained earlier in this section. 1 + i23 = 21cos 60° + i sin 60°2 -223 + 2i = 41cos 150° + i sin 150°2 If we multiply the trigonometric forms and use identities for the cosine and the sine of the sum of two angles, then the result is as follows. 321cos 60° + i sin 60°24341cos 150° + i sin 150°24 = 2 # 41cos 60° # cos 150° + i sin 60° # cos 150° + i cos 60° # sin 150° + i2 sin 60° # sin 150°2 = 831cos 60° # cos 150° - sin 60° # sin 150°2 + i1sin 60° # cos 150° + cos 60° # sin 150°24 = 83cos160° + 150°2 + i sin160° + 150°24 = 81cos 210° + i sin 210°2 Add. Notice the following. • The absolute value of the product, 8, is equal to the product of the absolute values of the factors, 2 # 4. • The argument of the product, 210°, is equal to the sum of the arguments of the factors, 60° + 150°. With the TI-84 Plus calculator in complex and degree modes, the MATH menu can be used to find the angle and the magnitude (absolute value) of a complex number. Multiply the absolute values and the binomials. i2 = -1; Factor out i. cos1A + B2 = cos A # cos B - sin A # sin B; sin1A + B2 = sin A # cos B + cos A # sin B
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