Question

if K=sin(pie/18)*sin(5*pie/18)*sin(7*pie/18)
then find value of k

Answer 1

Answer on Question #44177, Math, Trigonometry

if $K = \sin(pie/18) * \sin(5 * pie/18) * \sin(7 * pie/18)$

then find value of $k$

Solution.

$$\begin{aligned} K &= \sin \left(\frac {\pi}{18}\right) \sin \left(\frac {5 \pi}{18}\right) \sin \left(\frac {7 \pi}{18}\right) \\ &= \sin \left(\frac {\pi}{2} - \frac {8 \pi}{18}\right) \sin \left(\frac {\pi}{2} - \frac {4 \pi}{18}\right) \sin \left(\frac {\pi}{2} - \frac {2 \pi}{18}\right) = \cos \left(\frac {8 \pi}{18}\right) \cos \left(\frac {4 \pi}{18}\right) \cos \left(\frac {2 \pi}{18}\right) \\ &= \cos \left(\frac {2 \pi}{18}\right) \cos \left(\frac {4 \pi}{18}\right) \cos \left(\frac {8 \pi}{18}\right) = \frac {2 \sin \left(\frac {2 \pi}{18}\right) \cos \left(\frac {2 \pi}{18}\right) \cos \left(\frac {4 \pi}{18}\right) \cos \left(\frac {8 \pi}{18}\right)}{2 \sin \left(\frac {2 \pi}{18}\right)} \\ &= \frac {\sin \left(\frac {4 \pi}{18}\right) \cos \left(\frac {4 \pi}{18}\right) \cos \left(\frac {8 \pi}{18}\right)}{2 \sin \left(\frac {2 \pi}{18}\right)} = \frac {2 \sin \left(\frac {4 \pi}{18}\right) \cos \left(\frac {4 \pi}{18}\right) \cos \left(\frac {8 \pi}{18}\right)}{4 \sin \left(\frac {2 \pi}{18}\right)} \\ &= \frac {\sin \left(\frac {8 \pi}{18}\right) \cos \left(\frac {8 \pi}{18}\right)}{4 \sin \left(\frac {2 \pi}{18}\right)} = \frac {2 \sin \left(\frac {8 \pi}{18}\right) \cos \left(\frac {8 \pi}{18}\right)}{8 \sin \left(\frac {2 \pi}{18}\right)} = \frac {\sin \left(\frac {1 6 \pi}{1 8}\right)}{8 \sin \left(\frac {2 \pi}{1 8}\right)} = \frac {\sin \left(\pi - \frac {2 \pi}{1 8}\right)}{8 \sin \left(\frac {2 \pi}{1 8}\right)} \\ &= \frac {\sin \left(\frac {2 \pi}{1 8}\right)}{8 \sin \left(\frac {2 \pi}{1 8}\right)} = \frac {1}{8}. \end{aligned}$$

Answer. $K = \frac{1}{8}$

www.AssignmentExpert.com