To find the exact value of [tex]\(\sin 60^\circ\)[/tex], let's go through the steps involved in the calculation:
1. Angle Conversion:
We start with the given angle [tex]\(60^\circ\)[/tex]. For trigonometric calculations, it's common practice to convert angles from degrees to radians because trigonometric functions in many contexts, including calculator functions, use radians.
[tex]\[
\text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
Substituting [tex]\(60^\circ\)[/tex]:
[tex]\[
60^\circ \times \left( \frac{\pi}{180} \right) = \frac{\pi}{3} \text{ radians}
\][/tex]
2. Sine Calculation:
Using the sine function, we find the sine of the converted angle:
[tex]\[
\sin \left( \frac{\pi}{3} \right)
\][/tex]
3. Exact Value:
From trigonometric identities or special triangles (like the 30-60-90 triangle), we know that the sine of [tex]\(60^\circ\)[/tex] (or [tex]\(\frac{\pi}{3}\)[/tex] radians) is:
[tex]\[
\sin 60^\circ = \frac{\sqrt{3}}{2}
\][/tex]
4. Verification Against Given Options:
Now, we need to verify which of the given options corresponds to this exact value. The options are:
- [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\frac{1}{2}\)[/tex]
- 0
The correct and exact value of [tex]\(\sin 60^\circ\)[/tex] is:
[tex]\[
\frac{\sqrt{3}}{2}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{\sqrt{3}}{2}}
\][/tex]
