To write the given expression [tex]\(\sin \frac{\pi}{3} \cos \frac{\pi}{7} - \sin \frac{\pi}{7} \cos \frac{\pi}{3}\)[/tex] as the sine or cosine of an angle, we should use the angle addition or subtraction formulas for sine and cosine.
We start by using the sine subtraction formula:
[tex]\[
\sin(A - B) = \sin A \cos B - \cos A \sin B
\][/tex]
In this context, we compare our given expression with the sine subtraction formula. We identify [tex]\(A = \frac{\pi}{3}\)[/tex] and [tex]\(B = \frac{\pi}{7}\)[/tex]:
[tex]\[
\sin \frac{\pi}{3} \cos \frac{\pi}{7} - \sin \frac{\pi}{7} \cos \frac{\pi}{3}
\][/tex]
This matches the format of the sine difference identity, [tex]\(\sin(A - B) = \sin A \cos B - \cos A \sin B\)[/tex].
So, we can rewrite the given expression as:
[tex]\[
\sin \left(\frac{\pi}{3} - \frac{\pi}{7}\right)
\][/tex]
Now, we need to simplify the angle in the argument of the sine function:
[tex]\[
\frac{\pi}{3} - \frac{\pi}{7}
\][/tex]
To subtract these fractions, we need a common denominator. The least common multiple of [tex]\(3\)[/tex] and [tex]\(7\)[/tex] is [tex]\(21\)[/tex]:
[tex]\[
\frac{\pi}{3} = \frac{7\pi}{21}, \quad \frac{\pi}{7} = \frac{3\pi}{21}
\][/tex]
Now, subtract the fractions:
[tex]\[
\frac{7\pi}{21} - \frac{3\pi}{21} = \frac{4\pi}{21}
\][/tex]
Therefore, the expression [tex]\(\sin \frac{\pi}{3} \cos \frac{\pi}{7} - \sin \frac{\pi}{7} \cos \frac{\pi}{3}\)[/tex] can be written as:
[tex]\[
\sin \left(\frac{4\pi}{21}\right)
\][/tex]
So the expression is:
[tex]\[
\sin \left(\frac{4\pi}{21}\right)
\][/tex]
