(1 point) Solve the equation in the interval [0,2π]. If there is more than one solution write them separated by commas. (sin(x))2=1/36

Question

asked Mar 15, 2020 93.1k views

1 Answer

John Wiseman · Answer 1 · 2020-03-19T10:50:20+0000

Recall the double angle identity,

$\sin^2x=\frac{1-\cos2x}2$

Then

$\sin^2x=\frac{1-\cos2x}2=\frac1{36}\implies\cos2x=(17)/(18)$

$\cos x$ has a period of
$2\pi$ , so that
$\cos x=\cos(x+2n\pi)$ for any integer
. This means

$2x=\cos^(-1)(17)/(18)+2n\pi$

$\implies x=\frac12\cos^(-1)(17)/(18)+n\pi$

We get solutions of this form in the interval
$[0,2\pi]$ for
n=0 and
n=1 , giving

$x=\frac12\cos^(-1)(17)/(18)$

or

$x=\frac12\cos^(-1)(17)/(18)+\pi$