Problem 16 - Algebra and Trigonometry

Question

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. $$ \left(3, \frac{7 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The point with given polar coordinates \( (3, \frac{7 \pi}{6}) \) is plotted at a distance of 3 units from the pole 'O' (origin of the coordinate system), in the direction making an angle of 210 degrees with the positive x-axis.

Step 1: Identify given polar coordinates

The given polar coordinates are \( (3, \frac{7 \pi}{6}) \). This means that the point is at a distance of 3 units from the pole and at an angle of \( \frac{7 \pi}{6} \) measured in radian from the positive x-axis (the polar axis).

Step 2: Convert the polar angle from radians to degrees

While the polar system can deal with angles in both degrees and radians, plotting is sometimes easier in degrees. So, convert \( \theta \) to degree. We know that \( \pi \) radians = 180 degrees. Using this concept, \( \frac{7 \pi}{6} \) radians equals 210 degrees.

Step 3: Plotting the point

To plot this point, start from the pole 'O'. Measure an angle of 210 degrees in the anti-clockwise direction starting from the positive x-axis. From this position, move 3 units away from 'O' along the line drawn. This is the required point in the polar coordinate system.

Key Concepts

Polar Coordinate System

The polar coordinate system is an alternative to the rectangular (Cartesian) coordinate system for representing points on a two-dimensional plane. Unlike the rectangular system, which uses horizontal and vertical components (x and y coordinates), the polar system defines the location of a point based on its distance from a reference point and the angle from a reference direction.

In polar coordinates, a point is described by a pair \( (r, \theta) \), where \( r \) is the radius or the distance from the central reference point, called the pole, similar to the origin in the Cartesian system. The angle \( \theta \) is measured from the positive x-axis (also called the polar axis) and is typically given in radians or degrees.

When plotting points in polar coordinates, it's vital to remember that the angle can be positive or negative, indicating the direction in which you measure from the polar axis. A positive angle is measured counterclockwise, while a negative angle is measured clockwise. This unique system is particularly useful in fields that involve circular or rotational motion, such as physics and engineering.

Radian to Degree Conversion

Radian and degree are two units for measuring angles. While degrees are more familiar in everyday usage, radians are the preferred unit in mathematics and various scientific fields due to their natural connection with the circumference of a circle.

To convert an angle from radians to degrees, you can use the fact that a full circle measures \( 2\pi \) radians, which is equivalent to 360 degrees. Therefore, one radian equals \( \frac{180}{\pi} \) degrees.To convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \) as follows: \[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \. \]For example, to convert \( \frac{7\pi}{6} \) radians to degrees, you calculate: \[ \frac{7\pi}{6} \times \frac{180}{\pi} = 210 \text{ degrees}. \]This conversion is essential for those more comfortable with visualizing angles in degrees or when using devices like protractors, which typically measure in degrees.

Plotting Points in Polar Coordinates

Plotting points in polar coordinates involves two main steps: measuring the angle from the polar axis and moving the correct distance from the pole. This process can seem tricky at first, but it becomes more natural with practice. Here's how to do it:

Step 1: Start at the pole, the center of your polar coordinate system, which is analogous to the origin in a Cartesian grid.Step 2: Measure the angle \( \theta \) from the polar axis (positive x-axis) in the direction indicated by the sign of the angle. Remember, angles measured in the counterclockwise direction are positive, and those measured in the clockwise direction are negative.Step 3: From the direction defined by the angle, move outward a distance equal to \( r \) units. This distance is always non-negative, as it represents a magnitude.

For the given point \( (3, \frac{7\pi}{6}) \), you move counterclockwise from the polar axis to measure an angle of \( \frac{7\pi}{6} \) radians (or 210 degrees after conversion). From there, you measure outwards 3 units to arrive at the precise location of the point. Using polar coordinates makes it easy to plot points on circular paths and is integral to understanding complex numbers and sinusoidal functions.