How do you find the exact value of sin7.5?

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Answer 1 · 2015-05-16T07:02:01

Use some half angle formulas:

$sin(theta/2) = +-sqrt((1-cos theta) / 2)$

$cos(theta/2) = +-sqrt((1+cos theta) / 2)$

Also use a known value $cos 30^o = sqrt(3)/2$

If we stick to the first quadrant, we can take the sign of the square root to be $+$ in both cases.

$cos 15^o = sqrt((1+cos 30^o)/2)$

$= sqrt((1+sqrt(3)/2)/2)$

$= sqrt((2+sqrt(3))/4)$

$= sqrt(2+sqrt(3))/2$

$sin 7.5^o = sqrt((1-cos 15^o)/2)$

$= sqrt((1-sqrt(2+sqrt(3))/2)/2)$

$= sqrt((2-sqrt(2+sqrt(3)))/4)$

$= sqrt(2-sqrt(2+sqrt(3)))/2$