Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$

Question

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$

Answer 1

Notice, $\cos x$ has a period $2\pi$ hence, $\cos(2k \pi+\theta)=\cos \theta$ ($k$ is any integer) & $\cos(\pi+\theta)=-\cos\theta$,

Now, we have $$\cos\left(\frac{3\pi}{2}-t+2k \pi\right)=\cos\left(2k\pi+\left(\frac{3\pi}{2}-t\right)\right)$$ $$=\cos\left(\frac{3\pi}{2}-t\right)$$ $$=\cos\left(\pi+\left(\frac{\pi}{2}-t\right)\right)$$ $$=-\cos\left(\frac{\pi}{2}-t\right)$$ $$=-\sin(t)$$

Answer 2

Here is a geometrical interpretation :

The angle between the blue line and the $x$-axis denotes the angle $t$, whose $\sin$ is positive.

The cosine of $3π/2 - t$ (which is negative in this case) can be read thanks to the red line.

You can see that $\cos(3π/2 - t)=-\sin(t)$ may be true, looking at such a picture. This can be helpful then to prove your statement, as it is shown in the other answer.