In the image, you are asked to plot two points on a polar coordinate system:

1. $\left( 2, -\frac{5\pi}{4} \right)$
2. $\left( -6, \frac{\pi}{3} \right)$

Let's analyze each of these points to explain how they should be plotted.

1. $\left( 2, -\frac{5\pi}{4} \right)$

• r = 2: This tells you the distance from the origin.

• $\theta = -\frac{5\pi}{4}$: This is the angle in radians. A negative angle means that we rotate clockwise.

  • $-\frac{5\pi}{4}$ is equivalent to rotating 5π/4 radians clockwise, which is the same as $7\pi/4$ counterclockwise. This places the point in the third quadrant.

  Therefore, this point is plotted 2 units away from the origin, at an angle of $7\pi/4$ in the third quadrant.

2. $\left( -6, \frac{\pi}{3} \right)$

• r = -6: A negative radius means that the point is plotted in the opposite direction of the angle.

• $\theta = \frac{\pi}{3}$: This angle corresponds to 60 degrees counterclockwise, placing the point in the first quadrant.

  However, because the radius is negative, the point will be reflected in the opposite direction of $\frac{\pi}{3}$, which places it in the third quadrant, 6 units from the origin.

Plotting:

• The point $\left( 2, -\frac{5\pi}{4} \right)$ should be in the third quadrant, 2 units from the origin.
• The point $\left( -6, \frac{\pi}{3} \right)$ is also in the third quadrant but 6 units away from the origin.

From the drawing, the plot correctly reflects two points in the third quadrant, consistent with the above analysis.

Would you like more details or have any questions about plotting polar coordinates?

Here are 5 related questions for further practice:

1. How do you convert polar coordinates to Cartesian coordinates?
2. What is the effect of a negative radius in polar coordinates?
3. How would you convert an angle in radians to degrees in polar coordinates?
4. Can a polar coordinate system have multiple representations for the same point?
5. How would you plot the point $(-3, \pi)$ on a polar grid?

Tip: When working with negative radii, remember that the angle direction stays the same, but you reflect the point across the origin to the opposite quadrant.