Question

question 2 1 pts
what is the exact value of \\( \sin 75 ^ { \circ } \\)?
\\( \frac { \sqrt { 2 } - \sqrt { 6 } } { 4 } \\)
\\( \frac { - \sqrt { 6 } - \sqrt { 2 } } { 4 } \\)
\\( \frac { \sqrt { 2 } + \sqrt { 6 } } { 4 } \\)
\\( \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 } \\)

Explanation:

Step1: Split 75° into 45°+30°

$\sin75^\circ=\sin(45^\circ+30^\circ)$

Step2: Apply sine addition formula

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

Step3: Substitute values

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

Step4: Calculate the result

$\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}$

Answer:

$\frac{\sqrt{2}+\sqrt{6}}{4}$