use one or more of the six sum and difference identities to find the exact value of the expression. sin(60°…

Explanation:

Step1: Apply the sine sum identity

The sine sum identity is $\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here $A = 60^{\circ}$ and $B=45^{\circ}$. So, $\sin(60^{\circ}+45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}$.

Step2: Substitute the trigonometric values

We know that $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, $\cos45^{\circ}=\frac{\sqrt{2}}{2}$, $\cos60^{\circ}=\frac{1}{2}$, and $\sin45^{\circ}=\frac{\sqrt{2}}{2}$. Substituting these values, we get: [ \begin{align*} \sin(60^{\circ}+45^{\circ})&=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}\ &=\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4} \end{align*} ]

Step3: Combine the terms

Since the denominators are the same, we can combine the numerators: $\frac{\sqrt{6}+\sqrt{2}}{4}$.

Answer:

$\frac{\sqrt{6}+\sqrt{2}}{4}$