91Ó°ÊÓ

Plotting points using polar coordinates might seem confusing at first, but it becomes easier with practice.
In polar coordinates, points are represented as \( r, \theta \). \( r \) represents the distance from the origin (like the radius of a circle), and \( \theta \) represents the angle measured from the positive x-axis, going counterclockwise.
Here's how you can plot a point: Make sure you have polar graph paper, which consists of concentric circles for distance and lines radiating out from the center for angles. Find the angle first, then move outward along the circle for the distance.
For example, if you need to plot the point \( 3, 45^{\circ} \):

• Start at the origin (center of the polar graph).
• Find \( 45^{\circ} \) on the graph (halfway between the positive x-axis and the positive y-axis).
• Move 3 units outwards along the line radiating at \( 45^{\circ} \).

There you go, your point is plotted!

Understanding how to interpret a negative radius can be tricky.
In polar coordinates, a negative radius means you need to go in the opposite direction of the angle given.
Here's what you need to remember:

• Identify the angle first.
• Then move in the exact opposite direction of that angle.

Visualize this with an example, such as the point \( -3, 0^{\circ} \):
\( 0^{\circ} \) means facing directly along the positive x-axis, to the right. To interpret \( -3 \) radius, you should move left (opposite direction of the positive x-axis) by 3 units.
So, instead of moving to the right, you move to the left.

Angles in polar coordinates are measured in degrees going counterclockwise from the positive x-axis.
This is similar to how we measure angles in standard geometry. Here are essentials to remember:

• Zero degrees \( (0^{\circ}) \) points along the positive x-axis.
• Ninety degrees \( (90^{\circ}) \) points up along the positive y-axis.

Always measure angles counterclockwise unless specified otherwise. For instance, an angle of \( 180^{\circ} \) would point along the negative x-axis, directly opposite the positive x-axis.
Similarly, an angle of \( 270^{\circ} \) points down along the negative y-axis.
By understanding these basics, you can easily find the direction your point lies in and plot it accurately.