Question

If $0<\theta <2\pi$ and $2\cos \theta =\sqrt{3}\cos {10}^{\circ }-\sin {10}^{\circ }$, then the value of $\theta$ is

• A. ${20}^{\circ }$
• B. ${340}^{\circ }$
• C. ${320}^{\circ }$
• D. ${40}^{\circ }$

Answer 1

Answer: (C), (D)

${40}^{\circ }$

$2\cos \theta =\sqrt{3}\cos {10}^{\circ }-\sin {10}^{\circ }\Rightarrow \cos \theta =\frac{\sqrt{3}}{2}\cos {10}^{\circ }-\frac{1}{2}\sin {10}^{\circ }\Rightarrow \cos \theta =\cos {30}^{\circ }\cos {10}^{\circ }-\sin {30}^{\circ }\sin {10}^{\circ }\Rightarrow \cos \theta =\cos (30+10{)}^{\circ }=\cos {40}^{\circ }\Rightarrow \theta ={40}^{\circ }\text{or}{320}^{\circ }$