Question

The trigonometric equation $\sin x\cos x=2$ has
(a) One solution
(b) Two solutions
(c) Infinite solutions
(d) No solution

Answer 1

Hint: We will use the formula of double angle given by $\sin \left( 2x \right)=2\sin x\cos x$. With this, we will solve the trigonometric equation further. We will also apply $-1\le \sin \left( 2x \right)\le 1$ to check the value of it. If this is satisfied then we will find solutions for it. If not then we will have no solution.

Complete step-by-step solution -
We will consider the given trigonometric expression $\sin x\cos x=2$...(i)
Now we will divide and multiply on both the sides of the expression (i) by 2. Thus, we will get $\begin{align}
  & \dfrac{2}{2}\left( \sin x\cos x \right)=2\times \dfrac{2}{2} \\
 & \Rightarrow \dfrac{1}{2}\left( 2\sin x\cos x \right)=2 \\
\end{align}$
Now we use the trigonometric formula of sine function with double angle. The formula is given by $\sin \left( 2x \right)=2\sin x\cos x$. Thus, we will get $\dfrac{1}{2}\left( \sin \left( 2x \right) \right)=2$. Now we will take the 2 which is in the denominator of the left hand side to the right hand side of the equation. Thus, we get $\left( \sin \left( 2x \right) \right)=2\times 2$ which can also be written as $\sin \left( 2x \right)=4$.
As we know that the value of sine lies between – 1 and 1. So, we will use it here. We can write it numerically as $-1\le \sin \left( 2x \right)\le 1$. By substituting the value of $\sin \left( 2x \right)=4$ we get $-1\le 4\le 1$. As we know that the number 4 is greater than – 1 but is not smaller than 1. After this, we can clearly write that $-1\le 4\le 1$ is not possible. This means that $\sin \left( 2x \right)=4$ is not possible. Therefore, we have that the given trigonometric expression $\sin x\cos x=2$ has no solution.
Hence, the correct option is (d).

Note: Alternatively we can solve it with a different method. As we know that the maximum value of sine is 1. Therefore, we have here that the maximum value of $\sin \left( 2x \right)$ is also 1. So, clearly $\sin \left( 2x \right)=4$ is not possible at all as the value of $\sin \left( 2x \right)$ cannot be exceeded by 1. Therefore, we have that the given trigonometric expression $\sin x\cos x=2$ has no solution. Hence, the correct option is (d). With this trick, we can solve the question faster.