826 CHAPTER 8 Applications of Trigonometry The product obtained when multiplying by the first method is the rectangular form of the product obtained when multiplying by the second method. 81cos 210° + i sin 210°2 = 8a- 23 2 - 1 2 ib cos 210° = - 23 2 ; sin 210° = - 1 2 = -423 - 4i Rectangular form Product Theorem If r11cos u1 + i sin u12 and r21cos u2 + i sin u22 are any two complex numbers, then the following holds true. 3 r11cos U1 +i sin U12 4 # 3 r21cos U2 +i sin U22 4 =r1 r23 cos1U1 +U22 +i sin1U1 +U22 4 In compact form, this is written 1 r1 cis U12 1r2 cis U22 =r1 r2 cis1U1 +U22. That is, to multiply complex numbers in trigonometric form, multiply their absolute values and add their arguments. EXAMPLE 6 Using the ProductTheorem Find the product of 31cos 45° + i sin 45°2 and 21cos 135° + i sin 135°2. Write the answer in rectangular form. SOLUTION 331cos 45° + i sin 45°24321cos 135° + i sin 135°24 Write as a product. = 3 # 2 3cos145° + 135°2 + i sin145° + 135°24 Product theorem = 61cos 180° + i sin 180°2 Multiply and add. = 61-1 + i # 02 cos 180° = -1; sin 180° = 0 = -6 Rectangular form S Now Try Exercise 77. Quotients of Complex Numbers in Trigonometric Form The rectangular form of the quotient of 1 + i23 and -223 + 2i is found as follows. 1 + i23 -223 + 2i = A1 + i23 B A -223 - 2iB A -223 + 2iB A -223 - 2iB = -223 - 2i - 6i - 2i223 12 - 4i2 FOIL method; 1x + y21x - y2 = x2 - y2 = -8i 16 Simplify. = - 1 2 i Lowest terms Multiply both numerator and denominator by the conjugate of the denominator.
RkJQdWJsaXNoZXIy NjM5ODQ=