Answer:
\(\text{sin}(\frac{9\pi}{14})\).
Step-by-step explanation:
We have been given a trigonometric expression \(\text{sin}(\frac{\pi}{2})\text{cos}(\frac{\pi}{7})+\text{cos}(\frac{\pi}{2})\text{sin}(\frac{\pi}{7})\). We are asked to write our given expression as either the sine, cosine, or tangent of a single angle.
Using identity \(\text{sin}(a)\text{cos}(b)+\text{cos}(a)\text{sin}(b)=\text{sin}(a+b)\), we can rewrite our given expression.
Let \(a=\frac{\pi}{2}\) and \(b=\frac{\pi}{7}\).
Upon substituting these values in above identity, we will get:
\(\text{sin}(\frac{\pi}{2})\text{cos}(\frac{\pi}{7})+\text{cos}(\frac{\pi}{2})\text{sin}(\frac{\pi}{7})=\text{sin}(\frac{\pi}{2}+\frac{\pi}{7})\)
Upon simplifying right side of our equation, we will get:
\(\text{sin}(\frac{\pi}{2}+\frac{\pi}{7})=\text{sin}(\frac{\pi*7}{2*7}+\frac{\pi*2}{7*2})\)
\(\text{sin}(\frac{\pi}{2}+\frac{\pi}{7})=\text{sin}(\frac{7\pi}{14}+\frac{2\pi}{14})\)
\(\text{sin}(\frac{\pi}{2}+\frac{\pi}{7})=\text{sin}(\frac{7\pi+2\pi}{14})\)
\(\text{sin}(\frac{\pi}{2}+\frac{\pi}{7})=\text{sin}(\frac{9\pi}{14})\)
Therefore, our required expression would be \(\text{sin}(\frac{9\pi}{14})\).
