Problem 8 - College Trigonometry

Question

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(\frac{7}{2},-\frac{13 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The point can be expressed as \((-\frac{7}{2}, \frac{5\pi}{6})\), \((\frac{7}{2}, -\frac{\pi}{6})\), and \((\frac{7}{2}, \frac{23\pi}{6})\).

Step 1: Understanding the Given Coordinates

The point is given in polar coordinates as \( \left(\frac{7}{2}, -\frac{13 \pi}{6}\right) \). The first value \( \frac{7}{2} \) is the radial distance \( r \), and the second value \( -\frac{13 \pi}{6} \) is the angle \( \theta \) in radians.

Step 2: Converting Negative Angle to a Positive Angle

Since polar angles can loop every \( 2 \pi \), we can convert the negative angle to a positive one less than \( 2 \pi \). Add \( 2 \pi \) to \( -\frac{13 \pi}{6} \):\[-\frac{13 \pi}{6} + 2 \pi = -\frac{13 \pi}{6} + \frac{12 \pi}{6} = -\frac{\pi}{6} \] This new angle \( -\frac{\pi}{6} \) represents the same direction.

Step 3: Plotting the Given Point

Plot the point using the modified angle \( -\frac{\pi}{6} \) and radius \( r = \frac{7}{2} \). The angle \( -\frac{\pi}{6} \) means rotating clockwise by \( \frac{\pi}{6} \) from the positive \( x \)-axis, and marking the point at a distance \( \frac{7}{2} \) from the origin.

Step 4: Expressing the Point with \( r < 0 \)

To express the point with \( r < 0 \), change the direction using \( \theta = \theta + \pi \) while reversing the sign of \( r \):\[-\frac{\pi}{6} + \pi = \frac{5\pi}{6} \]Therefore, the coordinates become \( \left(-\frac{7}{2}, \frac{5\pi}{6}\right) \).

Step 5: Expressing the Point with \( r > 0 \) and \( \theta \leq 0 \)

Since the converted angle \( -\frac{\pi}{6} \) is already non-positive, use \( r > 0 \) directly. Thus, this expression remains \( \left(\frac{7}{2}, -\frac{\pi}{6}\right) \).

Step 6: Expressing the Point with \( r > 0 \) and \( \theta \geq 2 \pi \)

To fit the conditions, add \( 2 \pi \) to the positive equivalent angle \( \frac{11 \pi}{6} \):\[\frac{11 \pi}{6} + 2 \pi = \frac{11 \pi}{6} + \frac{12 \pi}{6} = \frac{23 \pi}{6} \]So the coordinates are \( \left(\frac{7}{2}, \frac{23\pi}{6}\right) \).

Key Concepts

Negative Angle Conversion

In polar coordinates, angles can be negative, which means they are measured in the clockwise direction from the positive x-axis. When working with negative angles, it can be helpful to convert them into positive angles, as it often simplifies understanding and plotting. This conversion relies on the property of angles that allows them to loop every full circle, or \(2\pi\) radians.
To convert a negative angle such as \(-\frac{13\pi}{6}\), you add \(2\pi\) radians:

• Start with \(-\frac{13\pi}{6}\)
• Add \(2\pi\) (which is equivalent to \(\frac{12\pi}{6}\))
• The result is \(-\frac{13\pi}{6} + \frac{12\pi}{6} = -\frac{\pi}{6}\)

The converted angle \(-\frac{\pi}{6}\) means the same direction as the original \(-\frac{13\pi}{6}\), but it's simpler to interpret.

Radial Distance

In polar coordinates, the radial distance \(r\) signifies how far away a point lies from the origin, centered at (0,0). The sign and the magnitude of \(r\) also provide information about the point's position.

• When \(r > 0\), the point is in the direction of \(\theta\) from the origin.
• When \(r < 0\), the point is in the opposite direction to \(\theta\).

This means if you are given a radial distance and wish to express it with a negative \(r\), you should also add or subtract \(\pi\) (half-turn) to the angle \(\theta\) to point to the exact opposite direction. For instance, to convert a point \(\left(\frac{7}{2}, -\frac{13\pi}{6}\right)\) into having \(r < 0\), the coordinates become \(\left(-\frac{7}{2}, \frac{5\pi}{6}\right)\). This adjustment maintains the point's location but flips the orientation.

Radian Angles

Radian is a unit of angular measure, defined such that a full circle is \(2\pi\) radians. Using radians provides a mathematical neatness in calculations involving angles due to its relationship with the mathematical constant \(\pi\).
The common fractions of \(\pi\) (such as \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}\), etc.) correspond to familiar angles in trigonometry, making them very useful.

• \(\theta = 0\) corresponds to the positive x-axis.
• \(\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2},...\) correspond to 30°, 60°, 90° angles respectively, when measuring counterclockwise from the x-axis.
• Negative angles measure clockwise.

Understanding radians simplifies the process of identifying equivalent angles, as with the exercise which showed that \(\frac{11\pi}{6}\) is equivalent to \(\frac{23\pi}{6}\) when adding a complete circle \(2\pi\) to it.

Plotting Polar Points

Plotting a point in polar coordinates revolves around the radial distance \(r\) and the angle \(\theta\). To plot a point like \(\left(\frac{7}{2}, -\frac{13\pi}{6}\right)\), follow these steps:

• Convert the negative angle to a positive equivalent, making it easier to visualize.
• Begin from the positive x-axis. Since \(-\frac{13\pi}{6}\) converts to \(-\frac{\pi}{6}\), you rotate clockwise \(\frac{\pi}{6}\).
• Move a distance of \(\frac{7}{2}\) units away from the origin along this line.

If converting to \(r < 0\), invert the direction by \(\pi\) (a half-turn), so the point \(\left(-\frac{7}{2}, \frac{5\pi}{6}\right)\) would sit \(\frac{7}{2}\) units in the line \(\theta = \frac{5\pi}{6}\), reflecting how the negative \(r\) flips the position across the origin.