Question

How do you determine the quadrant in which 6.02 radians lies.

Answer 1

Hint : The angles can have in the form of both degree and radians. The radians differ from the degree. Here in this question the radian value is given. By the value of quadrants in the radians used to determine the quadrant of the given radian value. Hence we obtain the required solution for the given question.

Complete step-by-step answer :
The value of an angle is represented in the form of degree and also in the form of radians. The radian value is differ from the value of degree but both represent the same.
The value of $\pi$ is 3.14radians.
Method 1:
As we know that there are four quadrants. The range of the values of the quadrant are given below:
First quadrant is from 0 to $\dfrac{\pi }{2} = 1.57$ radians.
The second quadrant is from 1.57 radians to $\pi = 3.14$ radians.
Third quadrant is from 3.14 radians to $\dfrac{{3\pi }}{2} = 4.71$ radians.
Fourth quadrant is from 4.71 radians to $2\pi = 6.28$ radians.
In the given question, we have 6.02 radians.
By the values of the quadrants the 6.02 radians lies in the fourth quadrant because the value in the fourth quadrant ranges from 4.71 radians to 6.28 radians. The value 6.02 radians come in between these numbers.
Hence the 6.02 radians lie in the fourth quadrant.
Method 2:
The radians is converted into degree by multiplying the given radian value to $\dfrac{{{{180}^ \circ }}}{{3.14}}$
So the value of 6.02 radians in degree is
$\Rightarrow 6.02 \times \dfrac{{{{180}^ \circ }}}{{3.14}}$
On simplifying we get
$\Rightarrow {345.095^ \circ }$
By rounding off the above number we have
$\Rightarrow {345.1^ \circ }$
First quadrant is from 0 to ${90^ \circ }$
The second quadrant is from ${90^ \circ }$ to ${180^ \circ }$ .
Third quadrant is from ${180^ \circ }$ to ${270^ \circ }$ .
Fourth quadrant is from ${270^ \circ }$ to ${360^ \circ }$ .
Therefore the ${345.1^ \circ }$ lies between the ${270^ \circ }$ and ${360^ \circ }$ .
Hence the ${345.1^ \circ }$ lies in the fourth quadrant.
So, the correct answer is “ fourth quadrant ”.

Note : The degree and radians are measurements for the angle. The values in degree and radians are different. While the value of $\pi$ is 180 degrees in the terms of degree and 3.14 radians in the form of radians. While converting the degree into radians or radians to the degree we must take care of the value of $\pi$ .