Question

Can anybody tell me what’s going on here?

Answer 1

Step-by-step explanation:

We are given a theta degree measure of sin in a specific interval. We are asked to find half of that degree measure.

The interval it ask us to find it in is between 2 pi and 5 pi over 2.

That is 360 degrees to 450 degrees. If you use reference angles, that is between the 0 degrees and 90 degrees. So that means our measure is going to be in the 1st quadrant.

Replace x with theta

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

We dont know cos x but we can use the pythagorean trig theorem to find cos x.

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

$\sin {}^{2} ( \frac{4}{5} ) + \cos {}^{2} (x) = 1$

$\frac{16}{25} + \cos {}^{2} (x) = 1$

$\cos {}^{2} (x) = 1 - \frac{16}{25}$

$\cos {}^{2} (x) = \frac{9}{25}$

$\cos(x) = \frac{3}{5}$

Cosine in 1st quadrant is positive so 3/5 is cos x.

Replace 3/5 for cos x.

$+ - \sqrt{ \frac{1 - \frac{3}{5} }{2} }$

$+ - \sqrt{ \frac{2}{10} }$

The answer is

$+ - \sqrt{ \frac{2}{10} }$