Evaluate $\cos (-2\pi)$ using radiants graph

Question

I plotted $2\pi$ on the $\cos$ graph and the result was $1$, so I assumed $-2\pi$ would be $-1$, but it turned out to be $1$.

Can someone explain how to work this out?

Answer 1

Yes, indeed: $\cos(-2\pi) = \cos(2\pi) = 1$. Formally, this is because $\cos \theta$ is an even function, meaning $\cos\theta = \cos(−\theta)$ for all $\theta$.

This can be seen in a number of ways. Try graphing $y = \cos x$ over the interval, say, $(-2\pi, 2\pi)$. For quick intuition, we can use, e.g., WolframAlpha:

Another factor to remember is that $\cos\theta$ is periodic, with period $2\pi$. So at any integer $k$ multiple of $2\pi$, $$\cos(2k\pi) = \cos(0) = \cos(2\pi) = 1$$ An easy way to convince yourself that this is the case is to recall that every integer multiple of $2\pi$ describes one or more full rotations about the unit circle. So $\cos(2k \pi) = \cos(0) = 1$.

Answer 2

$cos(x)=cos(-x)$ for every $x \in \mathbb R$, $cos(x) $ is an even function that why.