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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
on the negative imaginary axis, and a similar argument shows θ is coterminal with − π . The last
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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