Question

The value of $si{n}^2{5}^{\circ }+si{n}^2{10}^{\circ }+si{n}^2{15}^{\circ }$ + ..............+

$si{n}^2{85}^{\circ }+si{n}^2{90}^{\circ }$ is equal to

[Karnataka CET 1999]

• A. 7
• B. 8
• C. 9
• D.

Answer 1

The correct option is D

Given expression is

$si{n}^2{5}^{\circ }+si{n}^2{10}^{\circ }+si{n}^2{15}^{\circ }+..............+si{n}^2{85}^{\circ }+si{n}^2{90}^{\circ }$.

We know that $sin{90}^{\circ }=1orsi{n}^2{90}^{\circ }$ = 1.

Similarly, $sin{45}^{\circ }$ = $\frac{1}{\sqrt{2}}$ or $si{n}^2{45}^{\circ }$ = $\frac{1}{2}$ and the angles are in A.P. of 18 terms. We also know that

$si{n}^2{85}^{\circ }=\left[sin\right({90}^{\circ }-5^{\circ }){]}^2=co{s}^2{5}^{\circ }$.

Therefore from the complementary rule, we find $si{n}^2{5}^{\circ }+si{n}^2{85}^{\circ }=si{n}^2{5}^{\circ }+co{s}^2{5}^{\circ }$ = 1.

Therefore,

$si{n}^2{5}^{\circ }+si{n}^2{10}^{\circ }+si{n}^2{15}^{\circ }+.............+si{n}^2{85}^{\circ }+si{n}^2{90}^{\circ }$

= (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) + 1 + $\frac{1}{2}$ = 9 $\frac{1}{2}$.