Question

use a sum or difference formula to find the exact value of the trigonometric function.
\\( \sin 75 ^ { \circ } \\)
find the exact value of the expression.
\\( \sin 75 ^ { \circ } = \square \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. )

Explanation:

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

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

Step2: Use the sine sum formula

The sine sum formula 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 formula:

$$
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Answer:

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