Question

Given that sin x = 0.406 and that x is in quad 1, find sin (x/2)

Answer 1

\[ \sin(x) = 2 \sin\left ( \frac x 2 \right) \cos\left ( \frac x 2 \right)
\]

\[ .406 = 2 \sin\left ( \frac x 2 \right)\sqrt{ 1 - \sin^2\left ( \frac x 2 \right)}
\]
Solve for
\[ \sin\left ( \frac x 2 \right)
\]
to find
\[ \sin\left ( \frac x 2 \right)=.21, \sin\left ( \frac x 2 \right)=.98
\]
Since we are in the first quadrant, we should only take
\[ \sin\left ( \frac x 2 \right)=.21\]

Answer 2

what identity makes sin x = 2 sin (x/2) cos (x/2) ? i don't get that?

Answer 3

\[ \sin ( 2 a) = 2 \sin(a) \cos(b)\]
and the above comes from
\[ \sin(a+b) = \sin(a) \cos(b) + \sin(b) \cos(a) \]