Question

The equation$\sqrt{3}\sin x+\cos x=4$ has

• A. infinitely many solutions
• B. no solution
• C. two solution
• D. only one solution

Answer 1

The correct option is B

no solution

Simplify the given trigonometric equation:

The given equation is $\sqrt{3}\sin x+\cos x=4$.

Divide both sides of the equation by $2$.

$\begin{array}{rcl}\frac{\sqrt{3}}{2}\sin x+\frac{1}{2}\cos x& =& \frac{4}{2}\\ \frac{\sqrt{3}}{2}\sin x+\frac{1}{2}\cos x& =& 2\\ \sin x\cos 30°+\cos x\sin 30°& =& 2\\ \sin \left(x+30°\right)& =& 2\end{array}$

But range of sine function is $\left[-1,1\right]$.

So, it has no solution.

Hence, the equation has no solution and therefore option $\left(B\right)$ is correct.