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

• A. 7
• B. 8
• C. 9.5
• D. 10

Answer 1

The correct option is C

9.5

9.5

We have:

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

$=si{n}^2{5}^{\circ }+si{n}^2{10}^{\circ }+si{n}^2{15}^{\circ }+...+si{n}^2({90}^{\circ }-{10}^{\circ })+si{n}^2({90}^{\circ }-5^{\circ })+si{n}^2{90}^{\circ }$

$=si{n}^2{5}^{\circ }+si{n}^2{10}^{\circ }+si{n}^2{15}^{\circ }+...+co{s}^2{10}^{\circ }+co{s}^2{5}^{\circ }+si{n}^2{90}^{\circ }$

$=(si{n}^2{5}^{\circ }+co{s}^2{5}^{\circ })+(si{n}^2{10}^{\circ }+co{s}^2{10}^{\circ })+(si{n}^2{15}^{\circ }+co{s}^2{15}^{\circ })+(si{n}^2{20}^{\circ }+co{s}^2{20}^{\circ })+(si{n}^2{25}^{\circ }+co{s}^2{25}^{\circ })+(si{n}^2{30}^{\circ }+co{s}^2{30}^{\circ })+(si{n}^2{35}^{\circ }+co{s}^2{35}^{\circ })+(si{n}^2{40}^{\circ }+co{s}^2{40}^{\circ })+si{n}^2{45}^{\circ }+si{n}^2{90}^{\circ }$

$=1+1+1+1+1+1+1+1+\left(\frac{1}{\sqrt{2}}\right)+\left(1{)}^2\right[\because si{n}^2\theta +co{s}^2=1]$

$=8+\frac{1}{2}+1$

=9.5