Question

Find the value of ${\sin ^2}{5^ \circ } + {\sin ^2}{10^ \circ } + {\sin ^2}{15^ \circ } + ... + {\sin ^2}{90^ \circ }$
A) $\dfrac{{17}}{2}$
B) $\dfrac{{19}}{2}$
C) $\dfrac{{21}}{2}$
D) $\dfrac{{23}}{2}$

Answer 1

Hint:
At first we need to combine the terms in order in which we can write ${\sin ^2}({90^ \circ } - \theta ) = {\cos ^2}\theta $and by using the identity \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] we get the value of ${\sin ^2}{5^ \circ } + {\sin ^2}{10^ \circ } + {\sin ^2}{15^ \circ } + ... + {\sin ^2}{90^ \circ }$

Complete step by step solution:
 We can combine the given terms as follows
\[ \Rightarrow ({\sin ^2}{5^ \circ } + {\sin ^2}{85^ \circ }) + ({\sin ^2}{10^ \circ } + {\sin ^2}{80^ \circ }) + ({\sin ^2}{15^ \circ } + {\sin ^2}{75^ \circ }) + ... + ({\sin ^2}{40^ \circ } + {\sin ^2}{50^ \circ }) + {\sin ^2}{45^ \circ } + {\sin ^2}{90^ \circ }\]
Now $\sin ^{2} \theta$ can be written has ${\sin ^2}({90^ \circ } - \theta ) = {\cos ^2}\theta $
\[ \Rightarrow ({\sin ^2}{5^ \circ } + {\sin ^2}({90^ \circ } - 5)) + ({\sin ^2}{10^ \circ } + {\sin ^2}({90^ \circ } - 10)) + ... + ({\sin ^2}{40^ \circ } + {\sin ^2}({90^ \circ } - 40)) + {\sin ^2}{45^ \circ } + {\sin ^2}{90^ \circ }\]
\[ \Rightarrow ({\sin ^2}{5^ \circ } + {\cos ^2}{5^ \circ }) + ({\sin ^2}{10^ \circ } + {\cos ^2}{10^ \circ }) + ({\sin ^2}{15^ \circ } + {\cos ^2}{15^ \circ }) + ... + ({\sin ^2}{40^ \circ } + {\cos ^2}{40^ \circ }) + {\sin ^2}{45^ \circ } + {\sin ^2}{90^ \circ }\]
We know that \[{\sin ^2}\theta + {\cos ^2}\theta = 1\]
We get,
\[
   \Rightarrow 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + {\sin ^2}{45^ \circ } + {\sin ^2}{90^ \circ } \\
   \Rightarrow 8 + {\left( {\dfrac{1}{{\sqrt 2 }}} \right)^2} + 1 \\
   \Rightarrow 9 + \dfrac{1}{2} = \dfrac{{18 + 1}}{2} = \dfrac{{19}}{2} \\
\]

Therefore the correct option is (B).

Note:
An equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved.