sin(A + B) and sin(A - B)

The sine of a sum is not the sum of the sines. At 30° and 30°, $\sin\left(60^{\circ}\right)$ is about 0.87, while $\sin\left(30^{\circ}\right)+\sin\left(30^{\circ}\right)$ is 1. The true formulas mix sine and cosine:

$$\sin\left(A+B\right) = \sin\left(A\right)\cos\left(B\right)+\cos\left(A\right)\sin\left(B\right)$$

$$\sin\left(A-B\right) = \sin\left(A\right)\cos\left(B\right)-\cos\left(A\right)\sin\left(B\right)$$

They give exact values at angles the special triangles do not reach. 75° is 45° + 30°:

$$\sin\left(45^{\circ}+30^{\circ}\right) = \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2}$$

which is $\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}$, so $\sin\left(75^{\circ}\right) = \frac{\sqrt{6}+\sqrt{2}}{4}$, about 0.966. The sign in the middle follows the sign in the angle.