To rewrite the expression [tex]\(\sin \frac{\pi}{7} \cos x + \cos \frac{\pi}{7} \sin x\)[/tex] as the sine of an angle, we can make use of the sine addition formula. The sine addition formula is:
[tex]\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\][/tex]
Here, we should compare our expression [tex]\(\sin \frac{\pi}{7} \cos x + \cos \frac{\pi}{7} \sin x\)[/tex] with the right-hand side of the sine addition formula [tex]\(\sin A \cos B + \cos A \sin B\)[/tex].
By inspection, we can see:
- [tex]\(\sin \frac{\pi}{7}\)[/tex] corresponds to [tex]\(\sin A\)[/tex],
- [tex]\(\cos x\)[/tex] corresponds to [tex]\(\cos B\)[/tex],
- [tex]\(\cos \frac{\pi}{7}\)[/tex] corresponds to [tex]\(\cos A\)[/tex], and
- [tex]\(\sin x\)[/tex] corresponds to [tex]\(\sin B\)[/tex].
Thus, we can let [tex]\(A = \frac{\pi}{7}\)[/tex] and [tex]\(B = x\)[/tex].
Substituting these into the sine addition formula, we get:
[tex]\[
\sin \left(\frac{\pi}{7} + x\right) = \sin \frac{\pi}{7} \cos x + \cos \frac{\pi}{7} \sin x
\][/tex]
Therefore, the expression [tex]\(\sin \frac{\pi}{7} \cos x + \cos \frac{\pi}{7} \sin x\)[/tex] can be rewritten as:
[tex]\[
\sin \left(x + \frac{\pi}{7}\right)
\][/tex]
So the final rewritten expression is:
[tex]\[
\sin \left(x + \frac{\pi}{7}\right)
\][/tex]
