Question

The equation $\sqrt{3}\sin x+\cos x=4$ has
A) Infinitely many solutions
B) No solutions
C) Two solutions
D) Only one solution

Answer 1

For answering this question we should solve the given trigonometric expression $\sqrt{3}\sin x+\cos x=4$ . For doing that we will simplify and transform the given expression into the form of the basic trigonometric formulae $\sin A\cos B+\cos A\sin B=\sin \left( A+B \right)$.

Complete step by step solution:
Now considering from the question we have been asked to simplify the given trigonometric expression $\sqrt{3}\sin x+\cos x=4$ .
Now we will divide the whole expression by $2$ on both sides. By doing that we will have $\begin{aligned} & \dfrac{1}{2}\left( \sqrt{3}\sin x+\cos x \right)=\dfrac{4}{2} \\\ & \Rightarrow \dfrac{\sqrt{3}}{2}\sin x+\dfrac{1}{2}\cos x=2 \\\ \end{aligned}$
Now if we observe carefully then we can say that it is in the form of the basic trigonometric formulae $\sin A\cos B+\cos A\sin B=\sin \left( A+B \right)$ .
Now for further simplifying it we will use the values $\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}$ and $\sin {{30}^{\circ }}=\dfrac{1}{2}$ from the basic trigonometric concepts.
Now we can write this expression as $\Rightarrow \cos {{30}^{\circ }}\sin x+\sin {{30}^{\circ }}\cos x=2$ .
Now we will simplify and write it as $\Rightarrow \sin \left( {{30}^{\circ }}+x \right)=2$ .
From the basic concepts of trigonometry we know that the range of sine trigonometric function is given as $-1\le \sin \theta \le 1$ .
Since the value in the right hand side of the trigonometric expression is greater than one. So there does not exist any values for $x$ .
Hence we can conclude that the number of solutions of the given trigonometric expression $\sqrt{3}\sin x+\cos x=4$ is zero. So we need to mark the option $B$ as correct.
So, the correct answer is “Option B”.

Note: While answering questions of this type we should be sure with our trigonometric concepts. This question can be answered easily and in a short span of time and very few mistakes are possible in it. We have many other trigonometric formulae similarly for example some of them are $\sin A\cos B-\cos A\sin B=\sin \left( A-B \right)$ , $\cos A\cos B-\sin A\sin B=\cos \left( A+B \right)$ and $\cos A\cos B+\sin A\sin B=\cos \left( A-B \right)$ .