Question

In which quadrant does the terminal side of the angle $330^\circ$ lie?
A. Quadrant I
B. Quadrant II
C. Quadrant III
D. Quadrant IV

Answer 1

The Cartesian plane sweeps $360^\circ$ if we move from the x-axis at 1st quadrant in the anti-clockwise direction back to the x-axis at the first quadrant. So, we can say that the Cartesian plane is $360^\circ$. We know, there are $4$ quadrants. So, to find in which a given angle lies, we know, each quadrant sweeps $90^\circ$ of it’s own. That is, 1st quadrant sweeps $90^\circ$, 2nd quadrant sweeps $90^\circ$ and similar is the case with the 3rd and 4th quadrant. So, depending on the range of angle of the quadrants we can find the quadrant in which the given angle lies.

Complete step by step answer:
So, we first describe the ranges of all the four quadrants and then see in which quadrant does the angle $330^\circ$ lie.
The 1st quadrant ranges from $= 0^\circ$ to $90^\circ$
Now, the 2nd quadrant sweeps more$90^\circ$, so, to find the range of the 2nd quadrant, we will add $90^\circ$to the range of the 1st quadrant.
That is, the 2nd quadrant ranges from $= (0^\circ + 90^\circ )$ to $(90^\circ + 90^\circ )$= $90^\circ$ to $180^\circ$
Similarly, the 3rd quadrant sweeps more$90^\circ$, so, by adding $90^\circ$to the range of the 2nd quadrant we can find the range of the 3rd quadrant.

That is, the 3rd quadrant ranges from $= (90^\circ + 90^\circ )$ to $(180^\circ + 90^\circ )$= $180^\circ$ to $270^\circ$
Now, we can easily observe that the range of the 4th quadrant can be derived by adding $90^\circ$to the range of the 3rd quadrant.
That is, the 4th quadrant ranges from $= (180^\circ + 90^\circ )$ to $(270^\circ + 90^\circ )$ $= 270^\circ$to $360^\circ$
Now, the angle given to us in the problem is $330^\circ$.

So, we can see that it lies in the range of 4th quadrant as $270^\circ < 330^\circ < 360^\circ$.

Therefore, we can say that the given angle $330^\circ$ lies in the 4th quadrant.

Note: The Cartesian Plane can further extend infinitely to many more angles that exceed $360^\circ$. These angles are just repetitive and coincide with the angles lying between 1st and 4th quadrant. To find those angles we are to use the formula,
$\phi = n \times 360^\circ + \theta$
Where, $\phi$= the given angle which exceeds $360^\circ$
$\theta$= the angle with which $\phi$coincides in between 1st and 4th quadrant
$n$= $1,2,3,4....$