Question

How do you determine the quadrant in which $\dfrac{{9\pi }}{8}$ lies?

Answer 1

Remember the meaning of quadrant on a graph and how many quadrants are there in the graph also find the range of each of them in order to find in which quadrant the angle $\dfrac{{9\pi }}{8}$ will lie.

Complete step by step solution:
We will first know, what is a quadrant?

A coordinate system consists of two axes as well as four quadrants, yes a coordinate system consists of four quadrants which have their particular range of angles.

Let us see the range of all the four quadrants

The first quadrant has a range of $\left[ {0,\;\dfrac{\pi }{2}} \right]$ in radians and $[0,\;{90^ \circ }]$ in degrees.

The second quadrant has a range of $\left[ {\dfrac{\pi }{2},\;\pi } \right]$ in radians and $[{90^ \circ },\;{180^ \circ }]$ in degrees.

The third quadrant has a range of $\left[ {\pi ,\;\dfrac{{3\pi }}{2}} \right]$ in radians and $[{180^ \circ },\;{270^ \circ }]$ in degrees.

Last and the fourth quadrant have a range of $\left[ {\dfrac{{3\pi }}{2},\;2\pi } \right]$ in radians and $[{270^ \circ },\;{360^ \circ }]$ in degrees.
After seeing the ranges of quadrants, we get to know that each quadrant has an interval of width $\dfrac{\pi }{2}$ or ${90^ \circ }$

Now let us find on which quadrant $\dfrac{{9\pi }}{8}$ lies,

We can see that $\pi < \dfrac{{9\pi }}{8} < \dfrac{{3\pi }}{2}$, that means it is lying in the third quadrant.

Note: If the magnitude of an angle is greater than $2\pi$ then to find its quadrant we have to do

i. If the angle is positive: Divide that angle by $2\pi \;{\text{or}}\;{360^ \circ }$ depending upon in which unit the angle is. Note the remainder and then find the quadrant in which the remainder lies.

ii. If the angle is negative: Do the division similar to the positive angle and then subtract the remainder from $2\pi \;{\text{or}}\;{360^ \circ }$ accordingly. Then find the quadrant for the angle resulting from subtraction.