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**Unformatted text preview: **(z ) = 0
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and Im(z ) = 1. Since i is called the ‘imaginary unit,’ these answers make perfect sense.
2. To write a polar form of a complex number z , we need two pieces of information: the modulus
|z | and an argument (not necessarily the principal argument) of z . We shamelessly mine our
solution to Example 11.7.1 to ﬁnd what we need.
√
(a) For z = 3 − i, |z | = 2 and θ = − π , so z = 2cis − π . We can check our answer by
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√
converting it back to rectangular form to see that it simpliﬁes to z = 3 − i.
√
√
(b) For z = −2 + 4i, |z | = 2 5 and θ = π − arctan(2). Hence, z = 2 5cis(π − arctan(2)).
It is a good exercise to actually show that this polar form reduces to z = −2 + 4i.
(c) For z = 3i, |z | = 3 and θ = π . In this case, z = 3cis π . This can be checked
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geometrically. Head out 3 units from 0 along the positive real axis. Rotating π radians
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counter-clockwise lands you exactly 3 units above 0 on the imaginary axis at z = 3i.
(d) Last but not least, for z = −117, |z | = 117 and θ = π . We get z = 117cis(π ). As with
the previous problem, our answer is easily checked geometrically. 848 Applications of Trigonometry T...

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