To determine which of the given points has the polar coordinates [tex]\(\left(-5, \frac{2\pi}{3}\right)\)[/tex], let's break down what these polar coordinates mean:
1. Magnitude (-5): This represents the distance from the origin to the point. The negative sign indicates the direction is opposite in a straight line.
2. Angle [tex]\(\frac{2\pi}{3}\)[/tex]: This angle is measured from the positive x-axis counterclockwise to the line connecting the point to the origin.
Now, we need to identify the correct point among R, W, U, and Q that corresponds to these polar coordinates.
Given the polar coordinates:
- The negative magnitude means the point is in the direction opposite to the angle [tex]\(\frac{2\pi}{3}\)[/tex].
Let’s consider the angle [tex]\(\frac{2\pi}{3}\)[/tex]. This angle is in the second quadrant. Because we need to move in the opposite direction due to the negative radius, we effectively need to look in the direction of angle [tex]\(\pi + \frac{2\pi}{3} = \frac{5\pi}{3}\)[/tex].
- The angle [tex]\(\frac{5\pi}{3}\)[/tex] is equivalent to [tex]\(-\frac{\pi}{3}\)[/tex], which places our point in the fourth quadrant.
Among the points R, W, U, and Q, the one located in the fourth quadrant and has a distance of 5 units from the origin in the direction towards [tex]\( -\frac{\pi}{3} \)[/tex] will be our answer.
According to the steps to find this point, the answer is identified and matches with:
[tex]\[ Q \][/tex]
Thus, the point Q has the polar coordinates [tex]\(\left(-5, \frac{2\pi}{3}\right)\)[/tex].
