Question

The arithmetic mean of the numbers $2sin{2}^o,4sin{4}^o,6sin{6}^o,\dots \dots .178sin178^o,180sin180^o$ is

• A. $sin{1}^o$
• B. $cot{1}^o$
• C. $tan{1}^o$
• D. $cos{1}^o$

Answer 1

Answer: (B)

Total number of given numbers $=90$
Sum of numbers, $S=2\sin 2^{\circ }+4\sin 4^{\circ }+\dots \dots \dots +178\sin 178^{\circ }+180\sin 180^{\circ }$
$S=178\sin 2^{\circ }+176\sin 4^{\circ }+\dots \dots \dots +2\sin 178^{\circ }$
Adding both the equation, we get, $2S=180\left(\sin 2^{\circ }+\sin 4^{\circ }+\dots \dots \dots +\sin 178^{\circ }\right)+0$
$S=90\left(\sin 2^{\circ }+\sin 4^{\circ }+\dots \dots \dots +\sin 178^{\circ }\right)$
$\Rightarrow$ Mean $=\frac{S}{90}=\sin 2^{\circ }+\sin 4^{\circ }+\dots \dots +\sin 178^{\circ }$
$=\frac{\sin 89^{\circ }}{\sin 1^{\circ }}\sin \left(\frac{2^{\circ }+178^{\circ }}{2}\right)$
$=\frac{\cos 1^{\circ }}{{\sin }^{\circ }}\times \sin 90^{\circ }=\cot 1^{\circ }$