Trigonometric Functions — MATHS STD 11 Science — Question

3. 2

Solution:

We have:

$\sin^2\frac{\pi}{18}+\sin^2\frac{\pi}{9}+\sin^2\frac{7\pi}{2}+\sin^2\frac{4\pi}{9}$

$=\sin^2\frac{\pi}{18}+\sin^2\frac{2\pi}{8}+\sin^2\frac{7\pi}{18}+\sin^2\frac{8\pi}{2}$

$=\sin^2\frac{\pi}{18}+\sin^2\frac{2\pi}{8}+\sin^2\Big(\frac{7\pi}{18}\Big)+\sin^2\Big(\frac{8\pi}{2}\Big)$

$=\sin^2\frac{\pi}{18}+\sin^2\frac{2\pi}{18}+\sin^2\Big(\frac{\pi}{2}-\frac{\pi}{18}\Big)+\sin^2\Big(\frac{\pi}{2}-\frac{\pi}{18}\Big)$

$=\sin^2\frac{\pi}{18}+\sin^2\frac{2\pi}{18}+\cos^2\frac{2\pi}{18}+\cos^2\frac{\pi}{18}$

$=\sin^2\frac{\pi}{18}+\sin^2\frac{\pi}{18}+\cos^2\frac{2\pi}{18}+\cos^2\frac{2\pi}{18}$

$=1+1$

$=2$