Question

write the expression as the sine or cosine of an angle.\\
\\( \sin \frac { \pi } { 3 } \cos \frac { \pi } { 7 } - \sin \frac { \pi } { 7 } \cos \frac { \pi } { 3 } \\)\\
\\( \frac { \square \pi } { \square } \\)\\
hint: \\( \sin ( a \pm b ) = \sin a \cos b \pm \cos a \sin b \\)\\
\\( \cos ( a \pm b ) = \cos a \cos b \mp \sin a \sin b \\)

Explanation:

Step1: Recall the sine subtraction formula

The formula for \(\sin(A - B)=\sin A\cos B-\cos A\sin B\).

Step2: Identify \(A\) and \(B\)

In the given expression \(\sin\frac{\pi}{3}\cos\frac{\pi}{7}-\sin\frac{\pi}{7}\cos\frac{\pi}{3}\), we can see that \(A = \frac{\pi}{3}\) and \(B=\frac{\pi}{7}\).

Step3: Apply the formula

Substituting \(A\) and \(B\) into the formula \(\sin(A - B)\), we get \(\sin(\frac{\pi}{3}-\frac{\pi}{7})\).

Step4: Simplify the angle

Calculate \(\frac{\pi}{3}-\frac{\pi}{7}=\frac{7\pi - 3\pi}{21}=\frac{4\pi}{21}\).

Answer:

\(\sin\frac{4\pi}{21}\)