Evaluate the following: ^230^ + ^245^ + ^260^ + ^260^ + ^290^ (1) - Vidyadip

Question

Evaluate the following:
$\sin^230^\circ+\sin^245^\circ+\sin^260^\circ+\sin^260^\circ+\sin^290^\circ\dots(1)$

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Answer

$\sin^230^\circ+\sin^245^\circ+\sin^260^\circ+\sin^260^\circ+\sin^290^\circ\dots(1)$
By trigonometric ratios we have
$\sin30^\circ=\frac{1}{2}\ \ \ \ \sin45^\circ=\frac{1}{\sqrt2}$
$\sin60^\circ=\frac{\sqrt3}{2}\ \ \ \ \sin90^\circ=1$
By substituting above values in (i), we get
$=\Big[\frac{1}{2}\Big]^2+\Big[\frac{1}{\sqrt2}\Big]^2+\Big[\frac{\sqrt3}{2}\Big]^2+[1]^2$
$=\frac{1}{4}+\frac{1}{2}+\frac{3}{2}+1\Rightarrow\frac{1+3}{4}+\frac{1+2}{2}$
$\Rightarrow1+\frac{3}{2}=\frac{2+3}{2}=\frac{5}{2}$

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