Question

How do you find which quadrant each question is referring to if $\pi < a < \dfrac{{3\pi }}{2}$, $\dfrac{{3\pi }}{2} < b < 2\pi $?

Answer 1

Hint: In this question, we are given a range within which a particular angle lies and we have been asked the quadrant in which that angle will lie. Make a graph with all the four quadrants and mark their starting and ending angles. Then observe the given range and verify them with the quadrants. You will get your answer.

Complete step-by-step solution:
We have to find the quadrants these ranges are referring to. But first, we will know about each quadrant.
Quadrant 1: The angles lying in this quadrant ranges from $0^\circ $ to $90^\circ $. The x-axis and the y-axis, both are positive in this quadrant. So, we can say that the angles lying in this quadrant lie in $0 < x < \dfrac{\pi }{2}$.
Quadrant 2: The angles lying in this quadrant ranges from $90^\circ $ to $180^\circ $. The x-axis is negative and the y-axis is positive in this quadrant. So, we can say that the angles lying in this quadrant lie in $\dfrac{\pi }{2} < x < \pi $.
Quadrant 3: The angles lying in this quadrant ranges from $180^\circ $ to $270^\circ $. The x-axis and the y-axis, both are negative in this quadrant. So, we can say that the angles lying in this quadrant lie in $\pi < x < \dfrac{{3\pi }}{2}$.
Quadrant 4: The angles lying in this quadrant ranges from $270^\circ $ to $360^\circ $. The x-axis is positive and the y-axis is negative in this quadrant. So, we can say that the angles lying in this quadrant lie in $\dfrac{{3\pi }}{2} < x < 2\pi $.

Hence, as per the explanation given above, $\pi < a < \dfrac{{3\pi }}{2}$ is referring to quadrant 3, whereas $\dfrac{{3\pi }}{2} < b < 2\pi $ is referring to quadrant 4.

Note: How to convert degrees into radians?
We know that $\pi = 180^\circ $.
Then, what is the value of $90^\circ $, $270^\circ $ and $360^\circ $ in radians?
We will use a unitary method to find it out.
$ \Rightarrow \pi = 180^\circ $
$ \Rightarrow 1^\circ = \dfrac{\pi }{{180}}$
$ \Rightarrow $Therefore, $1^\circ \times 90^\circ = \dfrac{\pi }{{180}} \times 90^\circ $
We get, $90^\circ = \dfrac{\pi }{2}$
$ \Rightarrow $Similarly, $1^\circ \times 270^\circ = \dfrac{\pi }{{180}} \times 270^\circ $
On simplifying, we get $270^\circ = \dfrac{{3\pi }}{2}$
$ \Rightarrow $ Similarly, $1^\circ \times 360^\circ = \dfrac{\pi }{{180}} \times 360^\circ $
On simplifying, we get $360^\circ = 2\pi $
This is how degrees are converted into radians.