Problem 5 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(\theta=5\)

Step-by-Step Solution

Answer

Draw a line from the origin at an angle of 5 radians from the positive x-axis and extend it infinitely in both directions.

Step 1: Understand the Equation

The given equation is \(\theta = 5\). This is in polar coordinates, where \(\theta\) represents the angle from the positive x-axis.

Step 2: Identify the Angle

Identify where the angle \(\theta = 5\) radians is located. Note that in radians, \(\theta = 5\) is approximately 286 degrees (5 × 180/π).

Step 3: Draw the Reference Line

Draw a reference line from the origin making an angle \(\theta = 5\) radians with the positive x-axis. This line will be nearly in the fourth quadrant.

Step 4: Extend the Line

Extend this line in both directions from the origin infinitely, since the angle represents a direction, and there is no restriction on radius (r).

Step 5: Label the Graph

Label the angle on the graph as \(\theta = 5\). This helps to indicate that the line makes an angle of 5 radians with the positive x-axis.

Key Concepts

Graphing Polar Equations

Graphing polar equations might seem complicated at first, but breaking it down into steps makes it easier. To start, remember that in a polar coordinate system, each point is determined by a distance from the origin (\(r\)) and an angle (\(\theta\)) from the positive x-axis. For the equation \(\theta = 5\), we focus on the angle.

• First, identify the angle. \(\theta = 5\) means we are dealing with the angle \(5\) radians, not the distance.
• Next, recognize that this line passes through all points where the angle to the x-axis is exactly \(5\) radians.
• Instead of a single point, it forms a line extending infinitely in both directions from the origin at \(5\) radians.

By following these steps, you'll get your graph correctly. Practice with other angles to get comfortable with graphing polar equations.

Converting Radians to Degrees

Many polar coordinates are given in radians, but it can be useful to convert these into degrees for better understanding. The conversion between radians and degrees is straightforward: \(180\) degrees is equivalent to \(\pi\) radians. So, for any radian measure, you can convert it to degrees using the formula:
\(\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi}\).
Let's take our example of \(\theta = 5\) radians. To convert it:
\( \theta_{degrees} = 5 \times \frac{180}{\pi} \approx 286\) degrees.
This tells us the angle \(5\) radians forms an approximately \(286\) degrees angle with the positive x-axis.

Reference Angles

Understanding reference angles is key when working with polar coordinates. A reference angle is the smallest angle between the terminal side of the angle and the x-axis. Here's why they're helpful:
To find out where \(\theta = 5\) fits on a unit circle, we look at its reference angle:

• Subtract \(2\pi\) (one full rotation) from \(5\) since \(5\) radians is more than \(2\pi\) ( approximately \(6.28\)). \(5 - 2\pi \approx 5 - 6.28 \approx -1.28\) (negative, but helps us locate initial placement).
• Identify which quadrant it fits in. \(5\) radians places us around the fourth quadrant (as explained earlier).
• In simpler terms, a reference angle is finding equivalent smaller angles that give the position effectively.

Identification of reference angles makes graphing and conversion straightforward and helps in engaging with polar equations smoothly.