Trig and Radians Confusion

Question

I am learning about radians in my current class and am totally confused.

How does $\sin(x+\frac\pi 2)=\cos(x)$ when $\frac\pi 2<x$ < $\pi$.

I drew the triangles and I got

$\sin(x+\frac\pi 2)=\frac{-\mathrm{opposite}}{-\mathrm{hypotenuse}}=\frac{\mathrm{opp}}{\mathrm{hyp}}$

$\cos(x)=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$

If the adjacent equals the hypotenuse this should work however in most circumstances this isn't true. What is the major flaw in my work??

Here is what I drew out.

Answer 1

Your expression for $\sin(x)$ is correct. But recall that the cosine is the abscissa divided by the radius of the circle. So in your drawing, $\cos(x)=\frac a b$.

Answer 2

It's just how it is, if you look at a graph of sine and overlay cosine on top of it, then this would be readily apparent.