How do you find the exact values of sin 165° using the half angle formula?

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Answer 1 · 2015-08-30T13:29:31

$color(red)(sin(165) =sqrt(2–sqrt3)/2)$

The sine half-angle formula is

$sin(x/2) = ±sqrt((1 − cos x) / 2)$

The sign is positive if $x/2$ is in the first or second quadrant and negative if $x/2$ is in the third or fourth quadrant.

$sin165 = sin(330/2) = sqrt((1–cos 330)/2)$

$sin165= sqrt((1–cos(360-330)) / 2) = sqrt((1–cos30)/2)$

$sin165= sqrt((1–(sqrt3)/2)/2)= sqrt((2-sqrt3)/4)$

$sin165 =sqrt(2–sqrt3)/2$