Q.Using the compound angle formula, find the value of $\sin75^\circ$.

Since $75^\circ$ is not one of the standard table angles, it helps to write it as the sum of two angles that ARE in the table, then expand using the sine compound-angle formula.

Step 1 — split $75^\circ$ into standard angles. $75^\circ=45^\circ+30^\circ$.

Step 2 — apply the compound-angle formula. $\sin(A+B)=\sin A\cos B+\cos A\sin B$ with $A=45^\circ, B=30^\circ$:

Step 3 — substitute standard values. $\sin45^\circ=\cos45^\circ=\dfrac{\sqrt2}{2}$, $\cos30^\circ=\dfrac{\sqrt3}{2}$, $\sin30^\circ=\dfrac12$:

$\sin75^\circ=\dfrac{\sqrt2}{2}\cdot\dfrac{\sqrt3}{2}+\dfrac{\sqrt2}{2}\cdot\dfrac12=\dfrac{\sqrt6}{4}+\dfrac{\sqrt2}{4}=\dfrac{\sqrt6+\sqrt2}{4}$ …

Equivalently, split $75^\circ=30^\circ+45^\circ$ (swapping which angle is $A$ and which is $B$) — since addition is commutative in the formula, this gives t …

Using the formula for $\cos(A+B)$ (with the minus sign) instead of $\sin(A+B)$, or mixing up which term carries $\sin A\cos B$ versus $\cos A\ …