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

The correct option is C $\cos {10}^{\circ }$

$\sin {100}^{\circ }=\sin ({90}^{\circ }+{10}^{\circ })=\cos {10}^{\circ }$

$\sin {1000}^{\circ }=\sin ({360}^{\circ }\times 2+{280}^{\circ })=\sin {280}^{\circ }=\sin ({270}^{\circ }+{10}^{\circ }=-\cos {10}^{\circ }$

$\sin {10000}^{\circ }=\sin ({360}^{\circ }\times 27+{280}^{\circ })=\sin {280}^{\circ }=\sin ({270}^{\circ }+{10}^{\circ }=-\cos {10}^{\circ }$

$2\cos {10}^{\circ }+\sin {100}^{\circ }+\sin {1000}^{\circ }+\sin {10000}^{\circ }=2\cos {10}^{\circ }+\cos {10}^{\circ }-\cos {10}^{\circ }-\cos {10}^{\circ }=\cos {10}^{\circ }$