Question

use one or more of the six sum and difference identities to find the exact value of the expression.

\\( \sin \left( 75 ^ { \circ } \
ight) \\)

find the exact value of the expression.

\\( \sin \left( 75 ^ { \circ } \
ight) = \square \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

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

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

Step2: Use the sine sum identity \(\sin(A + B)=\sin A\cos B+\cos A\sin B\)

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:

$$
LATEXBLOCK0
$$

Answer:

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