Question

The number of principal solutions of $\tan 2\theta =1$ is
(a) One
(b) Two
(c) Three
(d) Four

Answer 1

Hint: To calculate the number of principal solutions of the given trigonometric equation, take inverse on both sides of the given equation. Find those values of $\theta$ which satisfy the given equation such that $0\le \theta \le 2\pi$. Rearrange the terms of the equation to calculate the number of solutions of the given equation.

Complete Step-by-step answer:
We have to calculate the number of principal solutions of $\tan 2\theta =1$. We observe that this is a trigonometric equation.
We know that principal solutions of a trigonometric equation include all the solutions lying in the range of $\left[ 0,2\pi \right]$.
So, we will calculate all the values of angle $\theta$ such that $0\le \theta \le 2\pi$.
Taking the inverse of $\tan 2\theta =1$ on both sides, we have $2\theta ={{\tan }^{-1}}1$.
Thus, we have $2\theta ={{\tan }^{-1}}1=\dfrac{\pi }{4},\dfrac{5\pi }{4}$.
Rearranging the terms of the above equation, we have $\theta =\dfrac{1}{2}\left( \dfrac{\pi }{4} \right),\dfrac{1}{2}\left( \dfrac{5\pi }{4} \right)=\dfrac{\pi }{8},\dfrac{5\pi }{8}$.
Hence, the number of principal solutions of the equation $\tan 2\theta =1$ is 2, which is option (b).

Note: We must keep in mind that we have to consider only principal solutions to the given trigonometric equation. If we will consider other solutions to the given equation, we will get an incorrect answer. We need to know the value of the expression ${{\tan }^{-1}}1$; otherwise, we won’t be able to solve the given question.