find the exact value of \\( \\sin 75 ^ { \\circ } \\) by using a sum or difference formula.

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 formula (\sin(A + B)=\sin A\cos B+\cos A\sin B)

Here (A = 45^{\circ}), (B=30^{\circ}). So (\sin75^{\circ}=\sin(45^{\circ}+30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ})

Step3: Substitute the values of trigonometric functions

We know that (\sin45^{\circ}=\frac{\sqrt{2}}{2}), (\cos30^{\circ}=\frac{\sqrt{3}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\sin30^{\circ}=\frac{1}{2}) [ \begin{align*} \sin75^{\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})