Question

The value of sin²5° + sin²10° + sin²15° + ... + sin²85° + sin²90° is

पर्याय

• 7

• 8

• 9.5

• 10

Answer 1

9.5

We have: 

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

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

$$={\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+...+{\cos }^2{10}^{\circ }+{\cos }^2{5}^{\circ }+{\sin }^2{90}^{\circ }$$

$$=({\sin }^2{5}^{\circ }+{\cos }^2{5}^{\circ })+({\sin }^2{10}^{\circ }+{\cos }^2{10}^{\circ })++({\sin }^2{15}^{\circ }+{\cos }^2{15}^{\circ })$$

$$+({\sin }^2{20}^{\circ }+{\cos }^2{20}^{\circ })+({\sin }^2{25}^{\circ }+{\cos }^2{25}^{\circ })+({\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ })$$

$$+({\sin }^2{35}^{\circ }+{\cos }^2{35}^{\circ })+({\sin }^2{40}^{\circ }+{\cos }^2{40}^{\circ })+{\sin }^2{45}^{\circ }+{\sin }^2{90}^{\circ }$$

$$=1+1+1+1+1+1+1+1+{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(1\right)}^2[\because {\sin }^2\theta +{\cos }^2\theta =1]$$

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

$$=9.5$$