Answer
a)
$\alpha =\frac{5\pi }{18}\text{ and }\beta =\frac{\pi }{9}$ in the expansion $\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}$.
b) The expression $\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}$ is equivalent to $\cos \frac{\pi }{6}$.
c) The exact value of $\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}$ is $\frac{\sqrt{3}}{2}$.
Work Step by Step
(a)
From the difference formula of cosines,
$\cos \left( \alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta $
The expansion using the above identity can be written as,
$\cos \left( \frac{5\pi }{18}-\frac{\pi }{9} \right)=\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}$
Compare the identity with the above expansion to determine the value of $\alpha \text{ and }\beta $.
Hence, $\alpha =\frac{5\pi }{18}\text{ and }\beta =\frac{\pi }{9}$.
(b)
Write the expansion using the cosine difference formula and solve as,
$\begin{align}
& \cos \left( \frac{5\pi }{18}-\frac{\pi }{9} \right)=\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9} \\
& \cos \frac{27\pi }{162}=\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9} \\
& \cos \frac{\pi }{6}=\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}
\end{align}$
Hence, the cosine of an angle $\frac{\pi }{6}$ is equivalent to the expression $\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}$.
(c)
Write the expansion using the cosine difference formula and solve as,
$\begin{align}
& \cos \left( \frac{5\pi }{18}-\frac{\pi }{9} \right)=\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9} \\
& \cos \frac{\pi }{6}=\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9} \\
& \frac{\sqrt{3}}{2}=\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}
\end{align}$
Hence, the exact value of $\cos \frac{5\pi }{18}\cos \frac{\pi }{9}+\sin \frac{5\pi }{18}\sin \frac{\pi }{9}$ is $\frac{\sqrt{3}}{2}$.
