863 CHAPTER 8 Test Prep Concepts Examples Trigonometric (Polar) Form of Complex Numbers Let the complex number x + yi correspond to the vector with direction angle u and magnitude r. x =r cos U y =r sin U r =!x2 +y2 tan U = y x , if x ≠0 The trigonometric (polar) form of the complex number x +yi is r 1cos U +i sin U2 or r cis U. Write 21cos 60° + i sin 60°2 in rectangular form. 21cos 60° + i sin 60°2 = 2 ¢ 1 2 + i # 23 2 ≤ = 1 + i23 Write -22 + i22 in trigonometric form. r = 4A -22 B 2 + A 22 B 2 = 2 tan u = -1 and u is in quadrant II, so u = 180° - 45° = 135°. -22 + i22 = 2 cis 135°. 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients Find the product and quotient, given z1 = 41cos 135° + i sin 135°2 and z2 = 21cos 45° + i sin 45°2. z1z2 = 81cos 180° + i sin 180°2 4 # 2 = 8; = 81-1 + i # 02 135° + 45° = 180° = -8 z1 z2 = 21cos 90° + i sin 90°2 4 2 = 2; = 210 + i # 12 135° - 45° = 90° = 2i Product and Quotient Theorems If r11cos u1 + i sin u12 and r21cos u2 + i sin u22 are any two complex numbers, then the following hold true. 3 r11cos U1 +i sin U12 4 # 3 r21cos U2 +i sin U22 4 =r1r2 3 cos1U1 +U22 +i sin1U1 +U22 4 and r11cos U1 +i sin U12 r21cos U2 +i sin U22 = r1 r2 3 cos1U1 −U22 +i sin1U1 −U22 4 , where r 2 1cos u2 + i sin u22 ≠0. Real Imaginary √2 0 u –√2 Find the dot product.82, 19 # 85, -29 = 2 # 5 + 11-22 = 8 Find the angle u between u = 83, 19 and v = 82, -39. cos u = u # v u v cos u = 3122 + 11-32 232 + 12 # 222 + 1-322 cos u = 32130 Simplify. cos u ≈0.26311741 Use a calculator. u ≈74.7° Dot Product The dot product of the vectors u = 8a, b9 and v = 8c, d9, denoted u # v, is given by the following. u # v =ac +bd Geometric Interpretation of the Dot Product If u is the angle between the two nonzero vectors u and v, where 0° … u … 180°, then the following holds true. cos U = u # v ∣ u∣ ∣ v∣ Geometric interpretation of the dot product Use the definitions. Use the inverse cosine function.
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