Question

How do you find which quadrant each question is referring to if $0<a<{\displaystyle \frac{\pi }{2}}$ and ${\displaystyle \frac{3\pi }{2}}<a<2\pi$?

Answer 1

Hint: We explain the central angle around a point. Then we discuss the algebraic and geometric versions of the quadrants. We also find different quadrants and their characteristics. Then we find the solutions for the angles $0<a<{\displaystyle \frac{\pi }{2}}$ and ${\displaystyle \frac{3\pi }{2}}<a<2\pi$.

Complete step by step answer:
The central angle around a point is always equal to ${360}^{\circ }$.
The total angle is divided into four parts or quadrant as they are called. These quadrants are named in roman numerals of $I,II,III,IV$.
In case of algebraic sense these quadrants give the signs of the x and y coordinates.
These quadrants are also called as the first, second, third and fourth quadrant.
The respective signs for the coordinates of $(x,y)$ will be $$(+,+),(-,+),(-,-),(+,-)$$ respectively for the quadrants.
Now we look for the geometric side of the quadrants where we deal with the angle of the trigonometric ratios.
The total circular angle of $2\pi$ can be divided into four parts. Each part is of ${\displaystyle \frac{\pi }{2}}$.
Therefore, the first quadrant is the interval of $(0,{\displaystyle \frac{\pi }{2}})$. The second quadrant is the interval of $({\displaystyle \frac{\pi }{2}},\pi )$. The third quadrant is the interval of $(\pi ,{\displaystyle \frac{3\pi }{2}})$. The fourth quadrant is the interval of $({\displaystyle \frac{3\pi }{2}},2\pi )$. The $0,{\displaystyle \frac{\pi }{2}},\pi ,{\displaystyle \frac{3\pi }{2}}$ indicates the axes.
Now we find for the quadrants $0<a<{\displaystyle \frac{\pi }{2}}$ and ${\displaystyle \frac{3\pi }{2}}<a<2\pi$.
When the angle $0<a<{\displaystyle \frac{\pi }{2}}$, the quadrant is first and when the angle ${\displaystyle \frac{3\pi }{2}}<a<2\pi$, the quadrant is fourth.

Note:
We can also represent the quadrants with respect to the image form of both algebraic and geometric versions.

The rotation of the coordinates happens anti-clockwise.