How do you evaluate $cos[ (5pi) / 8 ]$ ?

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How do you evaluate $cos[ (5pi) / 8 ]$ ?

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Answer 1 · 2016-01-08T13:53:30

$cos ((5pi)/8) = - (sqrt(2 - sqrt2))/2$

Explanation:

$cos ((5pi)/8) = cos ((-3pi)/8 + pi) = -cos ((3pi)/8)$
Find $cos ((3pi)/8)$ . Call cos x = $cos ((3pi)/8)$
$cos 2x = cos ((3pi)/4) = -sqrt2/2$
Apply the trig identity: $cos 2x = 2cos^2 x - 1$ . We get:
$2cos^2 x = 1 - sqrt2/2 = (2 - sqrt2)/2$
$cos^2 x = (2 - sqrt2)/4$
$cos x = cos ((3pi)/8) = +- (sqrt(2 - sqrt2))/2$
Finally,
$cos ((5pi)/8) = - cos ((3pi)/8) = - (sqrt(2 - sqrt2)/2)$

Check by calculator.
$cos ((5pi)/8) = cos 112^@5 = - 0.38$
$-(sqrt(2 - sqrt2)/2) = 0.768/2 = - 0.38$ . OK

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How do you evaluate $cos[ (5pi) / 8 ]$ ?

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