Question

The value of $2\cos 10^{\circ }+\sin 100^{\circ }+\sin 1000^{\circ }+\sin 10000^{\circ }$ is

• A. $0$
• B. $\sin 10^{\circ }$
• C. $\cos 10^{\circ }$
• D. $-1$

Answer 1

Answer: (C)

$2\cos 10^{\circ }+\sin 100^{\circ }+\sin 1000^{\circ }+\sin 10000^{\circ }$
$=2\cos 10^{\circ }+\sin \left(90^{\circ }+10^{\circ }\right)+\sin \left(3\times 360^{\circ }-80^{\circ }\right)+\sin \left(28\times 360^{\circ }-80^{\circ }\right)$
$=2\cos 10^{\circ }+\cos 10^{\circ }-\sin 80^{\circ }-\sin 80^{\circ }$
$=3\cos 10^{\circ }-2\cos 10^{\circ }$
$=\cos 10^{\circ }$