Question

Can anybody tell me what’s going on here?

Answer 1

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( (x)/(2) ) = + - \sqrt{ (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) ( (4)/(5) ) + \cos {}^(2) (x) = 1`

`(16)/(25) + \cos {}^(2) (x) = 1`

`\cos {}^(2) (x) = 1 - (16)/(25)`

`\cos {}^(2) (x) = (9)/(25)`

`\cos(x) = (3)/(5)`

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

Replace 3/5 for cos x.

`+ - \sqrt{ (1 - (3)/(5) )/(2) }`

`+ - \sqrt{ (2)/(10) }`

The answer is

`+ - \sqrt{ (2)/(10) }`