To find the exact value of [tex]\(\sin(345^\circ)\)[/tex], let's proceed step-by-step:
1. Recognize the reference angle:
[tex]\(345^\circ\)[/tex] is in the fourth quadrant. The reference angle is calculated by subtracting [tex]\(345^\circ\)[/tex] from [tex]\(360^\circ\)[/tex]:
[tex]\[
360^\circ - 345^\circ = 15^\circ
\][/tex]
So, the reference angle is [tex]\(15^\circ\)[/tex].
2. Use the sine value in the fourth quadrant:
In the fourth quadrant, the sine function is negative. Therefore:
[tex]\[
\sin(345^\circ) = -\sin(15^\circ)
\][/tex]
3. Find the exact value of [tex]\(\sin(15^\circ)\)[/tex]:
We use the angle subtraction formula for sine:
[tex]\[
\sin(15^\circ) = \sin(45^\circ - 30^\circ)
\][/tex]
Applying the sine difference formula [tex]\(\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)\)[/tex]:
[tex]\[
\sin(15^\circ) = \sin(45^\circ)\cos(30^\circ) - \cos(45^\circ)\sin(30^\circ)
\][/tex]
Using the known values of trigonometric functions:
[tex]\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2}
\][/tex]
Substituting these into the formula:
[tex]\[
\sin(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)
\][/tex]
Simplify the expression:
[tex]\[
\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
4. Determine the final expression for [tex]\(\sin(345^\circ)\)[/tex]:
Since [tex]\(\sin(345^\circ) = -\sin(15^\circ)\)[/tex]:
[tex]\[
\sin(345^\circ) = -\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) = -\frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
We can also rewrite this as:
[tex]\[
\sin(345^\circ) = \frac{-\sqrt{6} + \sqrt{2}}{4}
\][/tex]
So, the exact value of [tex]\(\sin(345^\circ)\)[/tex] is:
[tex]\[
\boxed{\frac{-\sqrt{6} + \sqrt{2}}{4}}
\][/tex]
