Pre-Calc Exam Notes 141

# Pre-Calc Exam Notes 141 - trigonometric form(sometimes...

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Complex Numbers Section 6.3 141 We know that any point ( x , y ) in the xy -coordinate plane that is a distance r > 0 from the origin has coordinates x = r cos θ and y = r sin θ , where θ is the angle in standard position as in Figure 6.3.1(a). x y 0 θ r ( x , y ) = ( r cos θ , r sin θ ) (a) Point ( x , y ) x y 0 θ r z = x + yi = r cos θ + ( r sin θ ) i (b) Complex number z = x + yi Figure 6.3.1 Let z = x + yi be a complex number. We can represent z as a point in the complex plane , where the horizontal x -axis represents the real part of z , and the vertical y -axis represents the pure imaginary part of z , as in Figure 6.3.1(b). The distance r from z to the origin is, by the Pythagorean Theorem, r = r x 2 + y 2 , which is just the modulus of z . And we see from Figure 6.3.1(b) that x = r cos θ and y = r sin θ , where θ is the angle formed by the positive x -axis and the line segment from the origin to z . We call this angle θ the argument of z . Thus, we get the
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Unformatted text preview: trigonometric form (sometimes called the polar form ) of the complex number z : For any complex number z = x + yi , we can write z = r (cos θ + i sin θ ) , where (6.3) r = | z | = R x 2 + y 2 and θ = the argument of z . The representation z = r (cos θ + i sin θ ) is often abbreviated as: z = r cis θ (6.4) In the special case z = = + i , the argument θ is unde±ned since r =| z | = 0. Also, note that the argument θ can be replaced by θ + 360 ◦ k or θ + π k , depending on whether you are using degrees or radians, respectively, for k = 0, ± 1, ± 2, . .. . Note also that for z = x + yi with r =| z | , θ must satisfy tan θ = y x , cos θ = x r , sin θ = y r ....
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