Question

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 \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), \(\sin30^{\circ}=\frac{1}{2}\)

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Answer:

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