Question

If the sine ratio is given, then how can sin 0/2 be determined?

Answer 1

Step-by-step explanation:

If you know, the sine. You must find the cos ratio, using the Pythagorean trig theorem.

Next, you use the half angle identity for sin

$\sin( \frac{ \alpha }{2} ) = \sqrt{ \frac{1 - \cos(a) }{2} }$

Example.

$\sin( \alpha ) = \frac{ \sqrt{3} }{2}$

We must find

$\sin( \frac{ \alpha }{2} )$

First, use the Pythagorean identity

$\sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) = 1$

$( \frac{ \sqrt{3} }{2} ) {}^{2} + \cos {}^{2} (a) = 1$

$\frac{3}{4} + \cos {}^{2} (a) = 1$

$\cos {}^{2} (a) = \frac{1}{4}$

$\cos( \alpha ) = \frac{1}{2}$

Now use the half angle identiy

$\sin( \frac{ \alpha }{2} ) = \sqrt{ \frac{1 - \cos( \alpha ) }{2} }$

$= \frac{ \sqrt{1 - \frac{1}{2} } }{ \sqrt{2} }$

$= \frac{ \sqrt{ \frac{1}{2} } }{ \sqrt{2} }$

$= \frac{ \sqrt{ \frac{1}{2} } }{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } = \frac{ \sqrt{1} }{ \sqrt{4} } = \frac{1}{2}$

So the answer for our example is 1/2