In which quadrant does the given angle of \[ - {514^ \circ }\] lie?

Hint: As we know that, there are two types of ways in which angles are measured. One is clockwise and the other is anticlockwise. As we know, a given angle is negative. Thus we will measure it in a clockwise direction. And then we will check the quadrant for the given angle.

Complete step by step answer:
Given the angle is \[ - {514^ \circ }\]
We will start measuring the angle from the positive x-axis in the clockwise direction.
After one full round the angle so completed will be \[ - {360^ \circ }\].
Now after adding more \[ - {90^ \circ }\] that is one more quadrant in clockwise direction, we get \[ - {360^ \circ } + ( - {90^ \circ }) = - {450^ \circ }\]
But this is not the given angle.
Again we will add \[ - {90^ \circ }\] that is one more quadrant in clockwise direction we get
\[ - {450^ \circ } + ( - {90^ \circ }) = - {540^ \circ }\]
But this is greater than the given angle.
Thus we conclude that the given angle lies in the third quadrant.

Note:
Note that, given angle is negative. So we will measure it in a clockwise direction. As we measured one full round and after that we will take the quadrant as per the requirement. But note that, when we measure the angle in anticlockwise direction, the same angle if taken positive the quadrant will be the second quadrant.