Turning by 90° and 180°

Adding a quarter or a half turn to an angle changes its sine and cosine in a fixed way, and the sum formulas show exactly how. The values of sine and cosine at 90°, 180° and 270° are 0, 1 or -1, so one product in each formula vanishes.

Take $\sin\left(x+180^{\circ}\right)$. The formula gives $\sin\left(x\right)\cos\left(180^{\circ}\right)+\cos\left(x\right)\sin\left(180^{\circ}\right)$. Put in $\cos\left(180^{\circ}\right) = -1$ and $\sin\left(180^{\circ}\right) = 0$: $\sin\left(x\right)\left(-1\right)+\cos\left(x\right)\left(0\right)$, which is $-\sin\left(x\right)$. Half a turn sends the point to the opposite side of the centre, so both coordinates change sign.

The cofunctions come out the same way: $\sin\left(90^{\circ}-x\right) = \sin\left(90^{\circ}\right)\cos\left(x\right)-\cos\left(90^{\circ}\right)\sin\left(x\right)$, which is cos x.