Question

What is the value of sin2 6° + sin2 12° + sin2 18° + … + sin2 84° + sin2 90°?​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{sin^2(6^{\circ})+sin^2(12^{\circ})+sin^2(18^{\circ})+\cdots+sin^2(84^{\circ})+sin^2(90^{\circ})}$

$\sf{=sin^2\left(\dfrac{\pi}{30}\right)+sin^2\left(\dfrac{2\pi}{30}\right)+sin^2\left(\dfrac{3\pi}{30}\right)+\cdots+sin^2\left(\dfrac{13\pi}{30}\right)+sin^2\left(\dfrac{14\pi}{30}\right)+sin^2\left(\dfrac{\pi}{2}\right)}$

$\sf{=\displaystyle\sum^{7}_{k=1}\,sin^2\left(\dfrac{k\,\pi}{30}\right)+\sum^{14}_{k=8}\,sin^2\left(\dfrac{k\,\pi}{30}\right)+1}$

$\sf{=\displaystyle\sum^{7}_{k=1}\,sin^2\left(\dfrac{k\,\pi}{30}\right)+\sum^{14}_{k=8}\,cos^2\left(\dfrac{\pi}{2}-\dfrac{k\,\pi}{30}\right)+1}$

$\sf{=\displaystyle\sum^{7}_{k=1}\,sin^2\left(\dfrac{k\,\pi}{30}\right)+\sum^{14}_{k=8}\,cos^2\left(\dfrac{15\pi-k\,\pi}{30}\right)+1}$

$\sf{=\displaystyle\sum^{7}_{k=1}\,sin^2\left(\dfrac{k\,\pi}{30}\right)+\sum^{14}_{k=8}\,cos^2\left\{\dfrac{(15-k)\pi}{30}\right\}+1}$

Put 15 - k = r

So,

$\sf{=\displaystyle\sum^{7}_{k=1}\,sin^2\left(\dfrac{k\,\pi}{30}\right)+\sum^{1}_{r=7}\,cos^2\left(\dfrac{r\,\pi}{30}\right)+1}$

$\sf{=\displaystyle\sum^{7}_{k=1}\,sin^2\left(\dfrac{k\,\pi}{30}\right)+\sum^{7}_{r=1}\,cos^2\left(\dfrac{r\,\pi}{30}\right)+1}$

$\sf{=\displaystyle\sum^{7}_{k=1}\left\{sin^2\left(\dfrac{k\,\pi}{30}\right)+cos^2\left(\dfrac{k\,\pi}{30}\right)\right\}+1}$

$\sf{=\displaystyle\sum^{7}_{k=1}\left\{1\right\}+1}$

$\sf{=7+1=8}$