How would you simplify
sin (2 pi + x)
and sin (-x)
thank you!
Also, a circle has $\displaystyle 2\pi$ radians in it, so any angle + $\displaystyle 2\pi$ equals that angle.
It's the same as adding 360 degrees, if you do a 360 you are right back where you started, angle x radians + $\displaystyle 2\pi$ radians = x radians
you can see this by choosing any x value you like, then typing into a calculator sin(x) and $\displaystyle sin(2\pi + x)$ (make sure your calculator is in radians and not in degrees)
some suggested values of x that you should try are $\displaystyle \frac{\pi}{6}, \frac{5}{6}, \pi$
i think the identity is proven using the unit circle but i just know it as the addition formula. but also, angel.white is correct, when we are talking about angles in the unit circle, we obtain equivalent angles by adding $\displaystyle 2 \pi$ to the angle, since we just make one complete revolution and come back to where we started.
so $\displaystyle \sin x = \sin (x + 2n \pi)$ for $\displaystyle n$ an integer