use a sum or difference formula to find the exact value of the trigonometric function. \n sin 165° \n sin…

Explanation:

Step1: Express (165^{\circ}) as a sum

Express (165^{\circ}) as (120^{\circ}+45^{\circ}). So, (\sin165^{\circ}=\sin(120^{\circ} + 45^{\circ})).

Step2: Apply the sum formula for sine

The sum formula for sine is (\sin(A + B)=\sin A\cos B+\cos A\sin B). Here (A = 120^{\circ}) and (B=45^{\circ}). We know that (\sin120^{\circ}=\frac{\sqrt{3}}{2}), (\cos120^{\circ}=-\frac{1}{2}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}), and (\cos45^{\circ}=\frac{\sqrt{2}}{2}). Substitute these values into the formula: [ \begin{align*} \sin(120^{\circ}+ 45^{\circ})&=\sin120^{\circ}\cos45^{\circ}+\cos120^{\circ}\sin45^{\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}\ &=\frac{\sqrt{6}-\sqrt{2}}{4} \end{align*} ]

Answer:

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