To find the exact value of [tex]\(\sin 165^\circ\)[/tex], we can use the identity involving trigonometric functions of complementary angles: [tex]\(\sin(180^\circ - x) = \sin x\)[/tex].
1. Understanding the Identity:
Since [tex]\(165^\circ\)[/tex] is close to [tex]\(180^\circ\)[/tex], we use:
[tex]\[
\sin(165^\circ) = \sin(180^\circ - 15^\circ) = \sin 15^\circ
\][/tex]
2. Using Angle Subtraction Identity for [tex]\( \sin 15^\circ \)[/tex]:
We can express [tex]\(15^\circ\)[/tex] as [tex]\(45^\circ - 30^\circ\)[/tex]. Using the sine angle subtraction identity [tex]\( \sin(A - B) = \sin A \cos B - \cos A \sin B \)[/tex]:
[tex]\[
\sin 15^\circ = \sin(45^\circ - 30^\circ)
\][/tex]
Therefore:
[tex]\[
\sin 15^\circ = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ
\][/tex]
3. Substitute the Known Values:
- [tex]\(\sin 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin 30^\circ = \frac{1}{2}\)[/tex]
Plugging these values 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]
[tex]\[
\sin 15^\circ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}
\][/tex]
Combine the terms:
[tex]\[
\sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
4. Verify Against Choices:
Comparing the calculated result to the options provided:
- [tex]\( \textbf{a.} \frac{\sqrt{2}+\sqrt{4}}{4} \)[/tex]
- [tex]\( \textbf{b.} \frac{\sqrt{2}-\sqrt{6}}{4} \)[/tex]
- [tex]\( \textbf{c.} \frac{\sqrt{6}+\sqrt{2}}{4} \)[/tex]
- [tex]\( \textbf{d.} \frac{\sqrt{6}-\sqrt{2}}{4} \)[/tex]
Therefore, the correct choice is:
[tex]\(\boxed{D}\)[/tex].
