Abdullah Knapp

2023-01-01

What is $\mathrm{cos}\left(-\theta \right)$ in terms of $\mathrm{cos}\theta$?

Steinunniqhd

Expert

Start from the point $\left(1,0\right)$ on the unit circle, which is when we have an angle of $0$ radians.
From there, try to move around the circumference while maintaining the same angle, first in a clockwise direction and then a counterclockwise direction. These two angles are $\theta$ and $-\theta$. As you can see, you'll end up with two points which lie on the same vertical line, which means that one is the reflection of the other with respect to the $x$-axis.
This means that the two points have coordinates $\left(x,y\right)$ and $\left(x,-y\right)$.
Since the cosine is the $x$-coordinate of the points on the unit circle, you see that the two points have the same cosine, and opposite sine.
In fact, the cosine is an even function, which means exactly that $\mathrm{cos}\left(x\right)=\mathrm{cos}\left(-x\right)$, while the sine is odd, which means that $\mathrm{sin}\left(x\right)=-\mathrm{sin}\left(-x\right)$.

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