Question

The value of $\theta$ satisfying $\cos \theta +\sqrt{3}\sin \theta =2$ is

• A. ${30}^{\circ }$
• B. ${60}^{\circ }$
• C. ${45}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is B

${60}^{\circ }$

Explanation for the correct option:

Find the required value.

Given: $\cos \theta +\sqrt{3}\sin \theta =2$

Dividing by $2$ on both sides we get

$$\begin{gathered} \Rightarrow \frac{1}{2}(\cos \theta +\sqrt{3}\sin \theta )=\frac{2}{2} \\ \Rightarrow \left(\frac{1}{2}\right)\cos \theta +\left(\frac{\sqrt{3}}{2}\right)\sin \theta =1 \\ \Rightarrow \sin {30}^{\circ }\cos \theta +\cos {30}^{\circ }\sin \theta =1 \\ \Rightarrow \sin ({30}^{\circ }+\theta )=\sin {90}^{\circ }\left[\because \sin (A+B)=\sin A\cos B+\cos A\sin B\right] \\ \Rightarrow {30}^{\circ }+\theta ={90}^{\circ } \\ \Rightarrow \theta ={90}^{\circ }-{30}^{\circ } \\ \Rightarrow \theta ={60}^{\circ } \end{gathered}$$

Hence, option (B) is the correct answer.