Question

The arithmetic mean of the numbers $2sin 2^{o},4sin ⁡ 4^{o},6sin ⁡ 6^{o},\ldots \ldots .178sin ⁡ 178^{o},180sin ⁡ 180^{o}$ 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, $S=2 \sin 2^{\circ}+4 \sin 4^{\circ}+\ldots \ldots \ldots+178 \sin 178^{\circ}+180 \sin 180^{\circ}$
$S=178 \sin 2^{\circ}+176 \sin 4^{\circ}+\ldots \ldots \ldots+2 \sin 178^{\circ}$
Adding both the equation, we get, $2 S=180\left(\sin 2^{\circ}+\sin 4^{\circ}+\ldots \ldots \ldots+\sin 178^{\circ}\right)+0$
$S=90\left(\sin 2^{\circ}+\sin 4^{\circ}+\ldots \ldots \ldots+\sin 178^{\circ}\right)$
$\Rightarrow $ Mean $=\frac{S}{90}=\sin 2^{\circ}+\sin 4^{\circ}+\ldots \ldots+\sin 178^{\circ}$
$=\frac{\sin 89^{\circ}}{\sin 1^{\circ}} \sin \left(\frac{2^{\circ}+178^{\circ}}{2}\right)$
$=\frac{\cos 1^{\circ}}{\sin ^{\circ}} \times \sin 90^{\circ}=\cot 1^{\circ}$