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Type: Multiple-Choice
Category: Complex Numbers
Level: Grade 11
Standards: HSN-CN.B.5
Author: nsharp1
Created: 5 years ago

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Complex Numbers Question

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For any

$z in CC$ , with

$r, theta in RR$ , and

$n$ a positive integer,

$z^n = r^n [cos(n theta) + i sin(n theta) ]$ .

Grade 11 Complex Numbers CCSS: HSN-CN.B.5

Which of the following gives the best reasoning of how, for

$n>3$ , this equation can be proved true?

There are more complicated trigonometric identities that deal with higher powers.
As was done in part b, by using the answer from the previous integer, each successive integer can be shown to be true using the same trigonometric identities (as used in part b).
Having shown it true for two different cases, it can be assumed true for all other cases.
For even powers of n, the process will be similar to $n=2$ , while for odd powers of n, the process of showing this equation is true will be similar to $n=3$ , but increasingly more complicated.