1. ## Simplify the following?

How would you simplify

sin (2 pi + x)

and sin (-x)

thank you!

2. Originally Posted by finalfantasy
How would you simplify

sin (2 pi + x)
use the addition formula for sine

$\sin (A + B) = \sin A \cos B + \sin B \cos A$

and sin (-x)
sine is an odd function, thus:

$\sin (-x) = - \sin x$

3. Thanks, would those be the answers using the unit circle?

4. Also, a circle has $2\pi$ radians in it, so any angle + $2\pi$ 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 + $2\pi$ radians = x radians

you can see this by choosing any x value you like, then typing into a calculator sin(x) and $sin(2\pi + x)$ (make sure your calculator is in radians and not in degrees)

some suggested values of x that you should try are $\frac{\pi}{6}, \frac{5}{6}, \pi$

5. Originally Posted by finalfantasy
Thanks, would those be the answers using the unit circle?
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 $2 \pi$ to the angle, since we just make one complete revolution and come back to where we started.

so $\sin x = \sin (x + 2n \pi)$ for $n$ an integer