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Question: If \(\cos\theta + \sqrt{3}\sin\theta = 2,\) then \(\theta =\) (only principal value)...

If cos⁡θ+3sin⁡θ=2,\cos\theta + \sqrt{3}\sin\theta = 2, then θ=\theta = (only principal value)

A

π3\frac{\pi}{3}

B

2π3\frac{2\pi}{3}

C

4π3\frac{4\pi}{3}

D

5π3\frac{5\pi}{3}

Answer

π3\frac{\pi}{3}

Explanation

Solution

12cos⁡θ+32sin⁡θ=22\frac{1}{2}\cos\theta + \frac{\sqrt{3}}{2}\sin\theta = \frac{2}{2} ⇒\Rightarrow cos⁡θcos⁡π3+sin⁡θsin⁡π3=1\mathbf{\cos}\mathbf{\theta}\mathbf{\cos}\frac{\mathbf{\pi}}{\mathbf{3}}\mathbf{+}\mathbf{\sin}\mathbf{\theta}\mathbf{\sin}\frac{\mathbf{\pi}}{\mathbf{3}}\mathbf{= 1}

⇒\Rightarrow cos⁡(θ−π3)=1=cos⁡0∘\mathbf{\cos}\left( \mathbf{\theta -}\frac{\mathbf{\pi}}{\mathbf{3}} \right)\mathbf{= 1 =}\mathbf{\cos}\mathbf{0}\mathbf{{^\circ}} ⇒θ−π3=0∘\mathbf{\Rightarrow \theta -}\frac{\mathbf{\pi}}{\mathbf{3}}\mathbf{= 0{^\circ}} ⇒\mathbf{\Rightarrow} θ=π3.\mathbf{\theta =}\frac{\mathbf{\pi}}{\mathbf{3}}\mathbf{.}