use one or more of the six sum and difference identities to find the\n\nsin(75°)\n\nfind the exact value of…

Explanation:

Step1: Express (75^{\circ}) as a sum of two known angles

We know that (75^{\circ}=45^{\circ} + 30^{\circ}).

Step2: Use the sine sum identity

The sine sum identity is (\sin(A + B)=\sin A\cos B+\cos A\sin B). Here (A = 45^{\circ}) and (B=30^{\circ}). We know that (\sin45^{\circ}=\frac{\sqrt{2}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\sin30^{\circ}=\frac{1}{2}), (\cos30^{\circ}=\frac{\sqrt{3}}{2}). Substitute these values into the identity: [ \begin{align*} \sin(45^{\circ}+ 30^{\circ})&=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\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})