Precalculus (6th Edition) Blitzer

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 $\frac{\sqrt{3}}{2}$.
(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 $\frac{\sqrt{3}}{2}$.