Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Plot each point, given its polar coordinates. Give two other pairs of polar coordinates for each point. Do not use a calculator. $$\left(3,120^{\circ}\right)$$

Step-by-Step Solution

Answer

The point \((3, 120^{\circ})\) can also be represented as \((3, 480^{\circ})\) and \((-3, 300^{\circ})\).

Step 1: Understand Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angular direction from the positive x-axis. For \((3, 120^{\circ})\), \(r = 3\) is the distance from the origin, and \(\theta = 120^{\circ}\) is the angle.

Step 2: Plot the Point

To plot the point \((3, 120^{\circ})\), start at the origin. Move outwards along a direction that forms a \(120^{\circ}\) angle from the positive x-axis. Place a point at the distance of 3 units along this line.

Step 3: Identify Other Polar Coordinates

To find other polar coordinates, remember each angle can have equivalents by adding or subtracting \(360^{\circ}\). So, one pair is \((3, 120^{\circ} + 360^{\circ}) = (3, 480^{\circ})\) and another is \((3, 120^{\circ} - 360^{\circ}) = (3, -240^{\circ})\).

Step 4: Use Negative Radius

Another approach is to use a negative radius with a different angle. A common equivalent for \((r, \theta)\) is \((-r, \theta + 180^{\circ})\). So, \((-3, 120^{\circ} + 180^{\circ}) = (-3, 300^{\circ})\) is also a valid representation.

Step 5: Verify New Coordinates

Check that each new pair corresponds to the same point. When plotting \((3, 480^{\circ})\), \((3, -240^{\circ})\), and \((-3, 300^{\circ})\), they all end up at the same location as the original \((3, 120^{\circ})\).

Key Concepts

Radial Distance

When discussing polar coordinates, one of the key components is the radial distance. This is the distance from the origin (commonly considered as the center point) to the specific point you are interested in. In the coordinate pair \((r, \theta)\), \(r\) denotes this radial distance.

• It dictates how far you "move out" from the center to plot your point.
• In our example, \((3, 120^{\circ})\), \(r = 3\), which means you'll be plotting the point 3 units away from the origin.

Understanding radial distance is crucial, as it helps you visualize positions on the plane by how far they "stretch" out from the origin. Imagine you are on a field, and the radial distance is like measuring out 3 meters from the center point before you make your turn (angle). While plotting points in polar coordinates, keeping the radial distance in mind helps in accurate placement on the graph.

Angular Direction

Angular direction is represented by \(\theta\) in polar coordinates and signifies the angle at which the point is located from the positive x-axis. This angle dictates the "direction" in which you place your point. Let's consider a few key aspects of angular direction:

• Starting from the positive x-axis, you measure angles counterclockwise.
• In the context of \((3, 120^{\circ})\), \(\theta = 120^{\circ}\) tells you to rotate counterclockwise by 120 degrees.
• If an angle is negative, it indicates a clockwise rotation.

By understanding angular direction, you can locate points effectively on the polar plane. It's similar to rotating a dial from a fixed point or direction. Knowing which way and how far to turn helps in determining the precise location of a point on the graph. When adjusting angles, always consider the reference line, the positive x-axis, to maintain consistency.

Equivalent Angles

Equivalent angles are all about finding different ways to reach the same spot in terms of angular direction. In polar coordinates, every angle can be represented in multiple ways by adding or subtracting 360 degrees, a full circle.

• For example, with our pair \((3, 120^{\circ})\), you can reach the same point with \((3, 480^{\circ})\) or \((3, -240^{\circ})\).
• This shows that while the angle value changes, the position of the point remains constant.

Another method for finding equivalent angles involves using a negative radius and adjusting the angle. You accomplish this by adding 180 degrees to the angle and using a negative radial distance, such as \((-3, 300^{\circ})\).Understanding equivalent angles not only affirms the flexibility of polar coordinates but also broadens your approach for plotting and interpreting points. It's like knowing that one path can loop around in different loops, yet still lead you to the same destination.