
Why is $x$ in Figure 1 $x$ and not $-x$? This has caused me to not understand why $\cos(-\theta) = x$ and $\cos(\theta) = x$.

Why is $x$ in Figure 1 $x$ and not $-x$? This has caused me to not understand why $\cos(-\theta) = x$ and $\cos(\theta) = x$.
The point has co-ordinates $(x,y)$. In the diagram, $x$ is a negative number, but that's OK - variables are allowed to stand for negative numbers. The angle $\theta$ is in the second quadrant, where the cosine function is negative, so $\cos\theta=x$ is perfectly consistent.