Solve the trigonometric equation $\sqrt{3}\cos \theta +\sin \theta =1$

Hint: The set of all the solutions of a given trigonometric equation constitute general solutions of the equation. In this question, the formula of the general solution for the given equation is $\sin \theta =\sin \alpha \Rightarrow \theta =n\pi +{{(-1)}^{n}}\alpha ,n\in Z$.

Complete step-by-step answer:

The equations that involve the trigonometric functions of a variable are called trigonometric equations. We will try to find the solutions of such equations. These equations have one or more trigonometric ratios of unknown angles. For example, $\cos x-\sin 2x=0$, is a trigonometric equation which does not satisfy all the values of x. Hence for such equations, we have to find the values of x or find the solution.

The given trigonometric equation is

$\sqrt{3}\cos \theta +\sin \theta =1$

Dividing both sides by 2, we get

$\dfrac{\sqrt{3}}{2}\cos \theta +\dfrac{1}{2}\sin \theta =\dfrac{1}{2}.......................(1)$

We know that, $\sin \left( \dfrac{\pi }{3} \right)=\dfrac{\sqrt{3}}{2},\cos \left( \dfrac{\pi }{3}

\right)=\dfrac{1}{2}$ and $\sin \left( \dfrac{\pi }{6} \right)=\dfrac{1}{2}$

Now put all the values in the equation (1), we get

$\sin \left( \dfrac{\pi }{3} \right)\cos \theta +\cos \left( \dfrac{\pi }{3} \right)\sin \theta =\sin

\left( \dfrac{\pi }{6} \right)$

Applying the sum formula of the sine trigonometric ratio of compound angles $\sin \left( A+B

\right)=\sin A\cos B+\cos A\sin B$ on the left side, we get

$\sin \left( \dfrac{\pi }{3}+\theta \right)=\sin \left( \dfrac{\pi }{6} \right)$

The general solution of the trigonometric equation \[\sin \theta =\sin \alpha \] is $\theta =n\pi +{{(-1)}^{n}}\alpha ,n\in Z$

$\dfrac{\pi }{3}+\theta =n\pi +{{(-1)}^{n}}\dfrac{\pi }{6}$, where $n\in Z$

Rearranging the terms, we get

$\theta =n\pi -\dfrac{\pi }{3}+{{(-1)}^{n}}\dfrac{\pi }{6}$, where $n\in Z$

Hence the general solution of the given trigonometric equation is $\theta =n\pi -\dfrac{\pi }{3}+{{(-1)}^{n}}\dfrac{\pi }{6}$, where $n\in Z$.

Note: Since all trigonometric ratios are periodic in nature, generally a trigonometric equation has more than one solution or an infinite number of solutions. There are basically three types of solutions- Particular solution, Principal Solution and General Solution.