Simplification of this trigonometric expression?

Question

Question: $\sin\bigl(\frac{5π}{2}-θ\bigr)$ How am I supposed to simplify this? I tried using the symmetry properties of the unit circle but still couldn't do it. Maybe something like $\sin(90 -θ) = \cos (θ)$?

Also for $\cos\bigl(\frac{5π}{2}+θ\bigr)$.

Answer 1

Yes, see where radius vector of angle terminates after rotation.

Remove multiples of $2 \pi; \dfrac{5\pi}{2} = 2\pi + \dfrac{\pi}{2},$ radius vector coterminal with $ \dfrac{\pi}{2}$.

So you are left with $\sin (\pi/2 -\theta) =\cos \theta$

Similarly next simplification is $ -\sin \theta$ with a negative sign as $\cos ..$ is negative in third quadrant, where the radius vector lands after a full rotation and a quarter.

Answer 2

HINT:

$$\frac{5\pi}{2} = 2\pi + \frac{\pi}{2}$$

and $\color{blue}{\cos}$ and $\color{blue}\sin$ are periodic with a period of $2\pi$