864 CHAPTER 8 Applications of Trigonometry Concepts Examples Let z = 41cos 180° + i sin 180°2. Find z3. 34 1cos 180° + i sin 180°243 Find z3. = 431cos 3 # 180° + i sin 3 # 180°2 = 641cos 540° + i sin 540°2 = 641-1 + i # 02 = -64 For z above, r = 4 and u = 180°. Find its square roots. 24 acos 180° 2 + i sin 180° 2 b = 210 + i # 12 = 2i 24 acos 180° + 360° 2 + i sin 180° + 360° 2 b = 210 + i1-122 = -2i Rectangular and Polar Coordinates If a point has rectangular coordinates 1x, y2 and polar coordinates 1r, u2, then these coordinates are related as follows. x =r cos U y =r sin U r2 =x2 +y2 tan U = y x , if x ≠0 De Moivre’s Theorem 3 r1cos U +i sin U2 4 n =rn1cos nU +i sin nU2 nth Root Theorem If n is any positive integer, r is a positive real number, and u is in degrees, then the nonzero complex number r 1cos u + i sin u2 has exactly n distinct nth roots, given by the following. n! r 1cos A +i sin A2, or n! r cis A, where A = U +360° # k n , k = 0, 1, 2, . . . , n - 1. If u is in radians, then A = U +2Pk n , k = 0, 1, 2, . . . , n - 1. Find the rectangular coordinates for the point 15, 60°2 in polar coordinates. x = 5 cos 60° = 5 a 1 2b = 5 2 y = 5 sin 60° = 5¢ 3 2 ≤ = 523 2 The rectangular coordinates are Q5 2 , 523 2 R. Find polar coordinates for 1-1, -12 in rectangular coordinates. r = 21-122 + 1-122 = 22 tan u = 1 and u is in quadrant III, so u = 225°. One pair of polar coordinates for 1-1, -12 is A 22, 225°B . Graph r = 4 cos 2u. 180° 0° 90° 270° 4 r = 4 cos 2U 8.6 De Moivre’sTheorem; Powers and Roots of Complex Numbers 8.7 Polar Equations and Graphs Polar Equations and Graphs r =a cos U r =a sin U f Circles r2 =a2 sin 2U r2 =a2 cos 2U f Lemniscates r =a tb sin U r =a tb cos U f Limaçons r =a sin nU r =a cos nU f Rose curves x y –1 u –1
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