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This question group is public and is used in 2 tests.

Author: nsharp1
No. Questions: 3
Created: Jul 19, 2019
Last Modified: 5 years ago

Integer Powers of i

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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

For

$n=2$ , show that the equation is true. Hint: use trigonometric identities.

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

For

$n=3$ , show that the equation is true. Hint: use trigonometric identities and the result from the previous answer.

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

$n>3$ , 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.