How do you solve $Cos 2 theta = cos^2 theta - 1/2$ ?

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How do you solve $Cos 2 theta = cos^2 theta - 1/2$ ?

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Answer 1 · 2016-05-26T04:53:30

$pi/4, (3pi)/4, (5pi)/4, (7pi)/4$

Explanation:

Replace in the equation cos 2x by $(2cos^2 x - 1)$ -->
$2cos^2 x - 1 = cos^2 x - 1/2$
$cos^2 x = 1 - 1/2 = 1/2$
$cos x = +- 1/sqrt2 = +- sqrt2/2$
Trig table and unit circle give -->
a. $cos x = sqrt2/2$ --> $x = +- pi/4$
b. $cos x = -sqrt2/2$ --> $x = +- (3pi)/4$
Answers for $(0, 2pi)$ -->
$pi/4, (3pi)/4, (5pi)/4, (7pi)/4$

Note. Arc $-pi/4$ is co-terminal to arc $(7pi)/4$
Arc $(-3pi)/4$ is co-terminal to arc $(5pi)/4$

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How do you solve $Cos 2 theta = cos^2 theta - 1/2$ ?

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How do you solve $Cos 2 theta = cos^2 theta - 1/2$ ?