**Unformatted text preview: **z . 11.7 Polar Form of Complex Numbers 847 Since there are inﬁnitely many choices for θ ∈ arg(z ), there inﬁnitely many polar forms for z , so
we used the indeﬁnite article ‘a’ in Deﬁnition 11.3. It is time for an example.
Example 11.7.2.
1. Find the rectangular form of the following complex numbers. Find Re(z ) and Im(z ).
(a) z = 4cis 2π
3 π
(b) z = 2cis − 34 (c) z = 3cis(0) (d) z = cis π
2 2. Use the results from Example 11.7.1 to ﬁnd a polar form of the following complex numbers.
(a) z = √ 3−i (b) z = −2 + 4i (c) z = 3i (d) z = −117 Solution.
1. The key to this problem is to write out cis(θ) as cos(θ) + i sin(θ).
π
π
π
(a) By deﬁnition, z = 4cis 23 = 4 cos 23 + i sin 23 . After some simplifying, we get
√
√
z = −2 + 2i 3, so that Re(z ) = −2 and Im(z ) = 2 3.
π
π
π
(b) Expanding, we get z = 2cis − 34 = 2 cos − 34 + i sin − 34
√
√
√
z = − 2 − i 2, so Re(z ) = − 2 = Im(z ). . From this, we ﬁnd (c) We get z = 3cis(0) = 3 [cos(0) + i sin(0)] = 3. Writing 3 = 3 + 0i, we get Re(z ) = 3 and
Im(z ) = 0, which makes sense seeing as 3 is a real number.
(d) Lastly, we have z = cis π = cos π + i sin π = i. Since i = 0 + 1i, we get Re...

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