Questions: Plot the points with polar coordinates (3, π/3) and (-5,-π/2). Give two alternative sets of coordinate pairs for both points.

Solution Steps

The polar coordinates \(\left(3, \frac{\pi}{3}\right)\) and \(\left(-5, -\frac{\pi}{2}\right)\) can be plotted as follows:

1. For \(\left(3, \frac{\pi}{3}\right)\):

   • The radius \(r = 3\).
   • The angle \(\theta = \frac{\pi}{3}\) (60 degrees).
   • This point lies 3 units away from the origin at an angle of \(\frac{\pi}{3}\) from the positive \(x\)-axis.

2. For \(\left(-5, -\frac{\pi}{2}\right)\):

   • The radius \(r = -5\).
   • The angle \(\theta = -\frac{\pi}{2}\) (-90 degrees).
   • A negative radius means the point is reflected across the origin. Thus, this point is equivalent to \(\left(5, \frac{\pi}{2}\right)\), which lies 5 units away from the origin at an angle of \(\frac{\pi}{2}\) (90 degrees) from the positive \(x\)-axis.

To find alternative coordinate pairs for \(\left(3, \frac{\pi}{3}\right)\), we can add or subtract multiples of \(2\pi\) to the angle or use a negative radius:

1. Adding \(2\pi\) to the angle: \[ \left(3, \frac{\pi}{3} + 2\pi\right) = \left(3, \frac{7\pi}{3}\right). \]

2. Using a negative radius and adding \(\pi\) to the angle: \[ \left(-3, \frac{\pi}{3} + \pi\right) = \left(-3, \frac{4\pi}{3}\right). \]

To find alternative coordinate pairs for \(\left(-5, -\frac{\pi}{2}\right)\), we can add or subtract multiples of \(2\pi\) to the angle or use a positive radius:

1. Adding \(2\pi\) to the angle: \[ \left(-5, -\frac{\pi}{2} + 2\pi\right) = \left(-5, \frac{3\pi}{2}\right). \]

2. Using a positive radius and adding \(\pi\) to the angle: \[ \left(5, -\frac{\pi}{2} + \pi\right) = \left(5, \frac{\pi}{2}\right). \]

Final Answer