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)

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