Question

How do you determine the quadrant in which $ \left( { - \dfrac{{11\pi }}{9}} \right) $ lies ?

Answer 1

Hint: In the given question, we are provided with an angle and we are required to find the quadrant in which the given angle lies. One must have some prior knowledge of the basics of coordinate geometry and radian measure of angle that will help us in finding the quadrant with ease.

Complete step by step solution:
Now, we are given the angle in radian measure as $ \left( { - \dfrac{{11\pi }}{9}} \right) $ .
So, to convert the radian measure to degree measure, we first multiply the radian measure by $ {\left( {\dfrac{{180}}{\pi }} \right)^ \circ } $ so as to get the angle in degree measure. Then to convert degree to minute, we multiply the degrees by $ 60' $ to get the result in minutes and to convert the minutes to seconds by multiplying $ 60'' $ to the given number which is in minute the resultant number will be in second.
Now, we can convert the same angle in degree measure as follows:
\[ \Rightarrow \left( { - \dfrac{{11\pi }}{9}} \right) \times {\left( {\dfrac{{180}}{\pi }} \right)^ \circ }\]
Cancelling the common factors in numerator and denominator, we get,
\[ \Rightarrow \left( { - \dfrac{{11}}{1}} \right) \times {\left( {20} \right)^ \circ }\]
Simplifying the calculations, we get,
\[ \Rightarrow - {220^ \circ }\]
Now, we know that each quadrant consists of a right angle or $ {90^ \circ } $ .
Also, we know that negative angle means that the angle is measured moving in an anti-clockwise direction from the positive side of x axis.
So, moving in anticlockwise direction from the positive side of the x axis, we first get the fourth quadrant involving angles from $ 0 $ to $ - {90^ \circ } $ .
Then, we get the third quadrant involving angles from $ - {90^ \circ } $ to $ - {180^ \circ } $ .
Then, we get the second quadrant involving angles from $ - {180^ \circ } $ to $ - {270^ \circ } $ .
Now, \[ - {220^ \circ }\] lies between $ - {180^ \circ } $ to $ - {270^ \circ } $ . So, we conclude that the angle $ \left( { - \dfrac{{11\pi }}{9}} \right) $ lies in the third quadrant.
So, the correct answer is “third quadrant”.

Note: For the conversion, we have followed the unitary method in which we first find the value of one unit and then multiply it by the desired number of units. We can also find the quadrant in which the given angle lies by simply adding $ {360^ \circ } $ and then moving in clockwise direction instead of moving in anticlockwise direction from the positive x axis.