Question

How do you prove `sin(pi+x) = - sinx`? Double Angle Question: 1.) Express sin x/2 in terms of cos x Thank you so much!?

Answer 1

I) `sin (x + Pi) = - sin x`. There are 2 ways to prove:
a. By the trig unit circle,.
Example: x = Pi/3 (Quadrant I). And `(x + pi)` is in Quadrant III.
Then `sin (x + pi) = - sin x`.
b. By trig identity sin (a + b) = sin a.cos b + sin a. sin b
`sin (x + pi) = sin x cos pi + sin pi.cos x`Because cos pi = -1 and sin pi = 0.`then`sin (x + Pi) = -sin x#

2) `cos x = 1 - 2sin^2 (x/2)`
`sin^2 (x/2) = (1 - cos x)/2`

`sin (x/2) = + (or -) sqr[[(1 - cos x)]/2]`