Difference between revisions of "1981 AHSME Problems/Problem 24"
Math piggy (talk | contribs) m (→Solution) |
Math piggy (talk | contribs) m (→Solution) |
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<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)</cmath> | <cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)</cmath> | ||
− | < | + | |
+ | <math>=\boxed{\textbf{2\cos(n\theta)}},</math> | ||
which gives the answer <math>\boxed{\textbf{D}}.</math> | which gives the answer <math>\boxed{\textbf{D}}.</math> |
Latest revision as of 15:24, 6 July 2021
Problem
If is a constant such that
and
, then for each positive integer
,
equals
Solution
Multiply both sides by and rearrange to
. Using the quadratic equation, we can solve for
. After some simplifying:
Substituting this expression in to the desired gives:
Using DeMoivre's Theorem:
Because is even and
is odd:
$=\boxed{\textbf{2\cos(n\theta)}},$ (Error compiling LaTeX. Unknown error_msg)
which gives the answer