To begin, let's start with the first quadrant in the unit circle:
It's easy to see why(due to complementary angles):
$$\cos\left(\frac{\pi}{2}-\theta\right) = \sin(\theta)$$ $$\sin\left(\frac{\pi}{2}-\theta\right) = \cos(\theta)$$
But what about Quadrant 2, 3 and 4, how do I visualise and prove these using the unit circle?
Q2: $$\sin\left(\frac{\pi}{2}+\theta\right) = ???$$ $$\cos\left(\frac{\pi}{2}+\theta\right) = ???$$
Q3:
$$\sin\left(\frac{3\pi}{2}-\theta\right) = ???$$ $$\cos\left(\frac{3\pi}{2}-\theta\right) = ???$$
Q4:
$$\sin\left(\frac{3\pi}{2}+\theta\right) = ???$$ $$\cos\left(\frac{3\pi}{2}+\theta\right) = ???$$
$\sin\left(\dfrac{\pi}{2} - \theta\right)$
to obtain $\sin\left(\dfrac{\pi}{2} - \theta\right)$. – N. F. Taussig Oct 01 '21 at 09:05