To determine which of the given options is equivalent to [tex]\(\sin(30^\circ)\)[/tex], we will evaluate the trigonometric functions for the specific angles involved.
1. Evaluate [tex]\(\sin(30^\circ)\)[/tex]:
[tex]\[
\sin(30^\circ) = 0.49999999999999994
\][/tex]
2. Evaluate [tex]\(\cos(90^\circ)\)[/tex]:
[tex]\[
\cos(90^\circ) = 6.123233995736766 \times 10^{-17}
\][/tex]
3. Evaluate [tex]\(\cos(30^\circ)\)[/tex]:
[tex]\[
\cos(30^\circ) = 0.8660254037844387
\][/tex]
4. Evaluate [tex]\(\sin(60^\circ)\)[/tex]:
[tex]\[
\sin(60^\circ) = 0.8660254037844386
\][/tex]
5. Evaluate [tex]\(\cos(60^\circ)\)[/tex]:
[tex]\[
\cos(60^\circ) = 0.5
\][/tex]
We see that:
- [tex]\(\cos(90^\circ) = 6.123233995736766 \times 10^{-17}\)[/tex], which is extremely close to 0.
- [tex]\(\cos(30^\circ) = 0.8660254037844387\)[/tex], which is not equal to [tex]\(\sin(30^\circ)\)[/tex].
- [tex]\(\sin(60^\circ) = 0.8660254037844386\)[/tex], which is not equal to [tex]\(\sin(30^\circ)\)[/tex].
- [tex]\(\cos(60^\circ) = 0.5\)[/tex], which is indeed very close to [tex]\(\sin(30^\circ)\)[/tex].
Therefore, the function that is equivalent to [tex]\(\sin(30^\circ)\)[/tex] is:
[tex]\[
\cos(60^\circ)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\text{D}}
\][/tex]
