Stitz-Zeager_College_Algebra_e-book

As in the previous problem our solution is trivial to

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Unformatted text preview: that θ is coterminal with π , and the result follows. If Re(z ) = 0 and Im(z ) < 0, then z lies 2 on the negative imaginary axis, and a similar argument shows θ is coterminal with − π . The last 2 property in the theorem was already discussed in the remarks following Definition 11.2. Our next goal is to completely marry the Geometry and the Algebra of the complex numbers. To that end, consider the figure below. Imaginary Axis (a, b) ←→ z = a + bi ←→ (r, θ) bi |z √ a2 + = | b2 = r θ ∈ arg(z ) 0 a Real Axis Polar coordinates, (r, θ) associated with z = a + bi with r ≥ 0. We know from Theorem 11.7 that a = r cos(θ) and b = r sin(θ). Making these substitutions for a and b gives z = a + bi = r cos(θ) + r sin(θ)i = r [cos(θ) + i sin(θ)]. The expression ‘cos(θ) + i sin(θ)’ is abbreviated cis(θ) so we can write z = rcis(θ). Since r = |z | and θ ∈ arg(z ), we get Definition 11.3. A Polar Form of a Complex Number: Suppose z is a complex number and θ ∈ arg(z ). The expression: |z |cis(θ) = |z | [cos(θ) + i sin(θ)] is called a polar form for...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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