Question

name: due fri 1. use the appropriate angle sum or difference formula to find the exact value of \\(\sin(15^{\circ})\\).

Explanation:

Step1: Express 15° as a difference

We know that \(15^{\circ}=45^{\circ} - 30^{\circ}\). So we can use the sine difference formula \(\sin(A - B)=\sin A\cos B-\cos A\sin B\), where \(A = 45^{\circ}\) and \(B = 30^{\circ}\).

Step2: Recall exact values

We know that \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), and \(\sin30^{\circ}=\frac{1}{2}\).

Step3: Substitute into the formula

Substitute these values into the formula \(\sin(45^{\circ}-30^{\circ})=\sin45^{\circ}\cos30^{\circ}-\cos45^{\circ}\sin30^{\circ}\).
\[

$$\begin{align*} \sin(45^{\circ}-30^{\circ})&=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}\\ &=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\\ &=\frac{\sqrt{6}-\sqrt{2}}{4} \end{align*}$$

\]

Answer:

\(\frac{\sqrt{6}-\sqrt{2}}{4}\)