Problem 30 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ \left(2,-\frac{5 \pi}{4}\right) $$

Step-by-Step Solution

Answer

Convert \-\frac{5\pi}{4}\ to \frac{3\pi}{4}\ and plot 2 units from the origin at 135°.

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 angle measured counterclockwise from the positive x-axis.

Step 2: Convert Angle to Positive

Since the angle given is negative, convert \(-\frac{5\pi}{4}\) to a positive angle by adding \(2\pi\): \(-\frac{5\pi}{4} + 2\pi = \frac{3\pi}{4}\).

Step 3: Plot the Radial Distance

From the origin, mark a point that is 2 units away along the direction of \(\frac{3\pi}{4}\), which is an angle of 135°.

Step 4: Mark the Point

Plot the point at the intersection of a circle with radius 2 units and the line at angle \(\frac{3\pi}{4}\).

Key Concepts

Convert Angle

Angles in polar coordinates can be tricky, especially when they are negative. A negative angle essentially means that you are measuring the angle clockwise from the positive x-axis.
To simplify, you can convert a negative angle to a positive one by adding a full rotation, which is represented by \(2\pi\).
For example, \( -\frac{5\pi}{4} \) can be converted by adding \(2\pi\): \[ -\frac{5\pi}{4} + 2\pi = \frac{3\pi}{4} \]
This makes it easier to visualize and plot the point.
So, remember, if you have a negative angle, simply add \(2\pi\) to make it positive!

Radial Distance

The radial distance, or radius, is the distance from the origin (zero point) to the location of your point on the plane. In polar coordinates, this is the first number in the pair \(r, \theta\).
For example, in the point \(2, -\frac{5\pi}{4} \), the radial distance is \( r = 2 \).
This means you will measure 2 units from the origin along the direction of your angle.
If the radius is positive, you simply count that many units from the origin. If it's negative, you would measure the distance in the opposite direction of your angle.
In our example, we have a positive radius of 2, so we measure 2 units outwards along the line at the angle we converted earlier.

Plotting Points

Plotting points in polar coordinates involves both the angle and the radial distance. After converting your angle and noting your radial distance, you're ready to plot the point.
Here's the step-by-step:

• Start from the origin (0, 0).
• Move along the direction of your converted angle.
  For \(\frac{3\pi}{4} \), this means moving to 135° counterclockwise from the positive x-axis.
• Measure the radial distance along this line.
• In this case, you move 2 units along the 135° direction from the origin.

Finally, mark the point where you land after measuring the radial distance.
This point is represented by your original polar coordinates: (2, \( -\frac{5\pi}{4} \)). Using these steps ensures accurate plotting and helps you visualize points easily in polar coordinates.