To simplify the expression [tex]\( 2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} \)[/tex] using a double-angle formula, we can follow these steps:
1. Recall the Double-Angle Formula for Sine:
The double-angle formula for sine states that:
[tex]\[
\sin(2a) = 2 \sin(a) \cos(a)
\][/tex]
2. Identify the Given Expression:
We are given the expression [tex]\( 2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} \)[/tex].
3. Match the Given Expression to the Double-Angle Formula:
Notice that the given expression [tex]\( 2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} \)[/tex] matches the right-hand side of the double-angle formula [tex]\( \sin(2a) = 2 \sin(a) \cos(a) \)[/tex], where [tex]\( a = \frac{\pi}{7} \)[/tex].
4. Simplify Using the Double-Angle Formula:
Based on the formula, we can write:
[tex]\[
2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} = \sin \left( 2 \cdot \frac{\pi}{7} \right)
\][/tex]
Simplify the argument of the sine function:
[tex]\[
2 \cdot \frac{\pi}{7} = \frac{2\pi}{7}
\][/tex]
5. Final Simplified Expression:
Therefore, the simplified form of the given expression is:
[tex]\[
\sin \frac{2\pi}{7}
\][/tex]
6. Evaluate the Simplified Expression:
When evaluated, the value of [tex]\( \sin \frac{2\pi}{7} \)[/tex] is approximately:
[tex]\[
0.7818314824680298
\][/tex]
Hence, the simplified expression [tex]\( 2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} \)[/tex] equals [tex]\( \sin \frac{2\pi}{7} \)[/tex], and its numerical value is approximately [tex]\( 0.7818314824680298 \)[/tex].
