To rewrite [tex]\(\sin \left(\frac{5 \pi}{12}\right)\)[/tex] in terms of its cofunction, we can use the cofunction identity for sine. The cofunction identity is:
[tex]\[
\sin(x) = \cos\left(\frac{\pi}{2} - x\right)
\][/tex]
We will apply this identity to the given angle [tex]\(\frac{5 \pi}{12}\)[/tex].
1. Identify [tex]\(x\)[/tex]:
[tex]\[
x = \frac{5 \pi}{12}
\][/tex]
2. Apply the cofunction identity:
[tex]\[
\sin \left(\frac{5 \pi}{12}\right) = \cos \left(\frac{\pi}{2} - \frac{5 \pi}{12}\right)
\][/tex]
3. Simplify the expression inside the cosine function:
[tex]\[
\frac{\pi}{2} = \frac{6 \pi}{12} \text{ (since } \frac{\pi}{2} = \frac{6 \pi}{12} \text{)}
\][/tex]
[tex]\[
\cos \left(\frac{\pi}{2} - \frac{5 \pi}{12}\right) = \cos \left(\frac{6 \pi}{12} - \frac{5 \pi}{12}\right)
\][/tex]
4. Perform the subtraction:
[tex]\[
\frac{6 \pi}{12} - \frac{5 \pi}{12} = \frac{\pi}{12}
\][/tex]
Therefore:
[tex]\[
\sin \left(\frac{5 \pi}{12}\right) = \cos \left(\frac{\pi}{12}\right)
\][/tex]
Now, evaluating [tex]\(\cos \left(\frac{\pi}{12}\right)\)[/tex], we find that its value is approximately:
[tex]\[
\cos \left(\frac{\pi}{12}\right) \approx 0.9659258262890683
\][/tex]
Hence, the function rewritten in terms of its cofunction and evaluated is:
[tex]\[
\boxed{0.9659258262890683}
\][/tex]
