Problem 23 - Precalculus : Building Concepts and Connections

Question

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(-4, \frac{7 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The rectangular coordinates equivalent to the given polar coordinates \( \left(-4, \frac{7 \pi}{6}\right) \) are \( (2, -2\sqrt{3}) \).

Step 1: Understanding the Formulas

The formulas used to convert polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \) are: \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \). The given polar coordinates are \( r = -4 \) and \( \theta = \frac{7 \pi}{6} \).

Step 2: Calculating the x-coordinate

To calculate the x-coordinate, use the formula \( x = r \cdot \cos(\theta) \). Substituting the given values, \( x = -4 \cdot \cos(\frac{7 \pi}{6}) \). This simplifies to \( x = 2 \).

Step 3: Calculating the y-coordinate

To calculate the y-coordinate, use the formula \( y = r \cdot \sin(\theta) \). Substituting the given values, \( y = -4 \cdot \sin(\frac{7 \pi}{6}) \). This simplifies to \( y = -2\sqrt{3} \).

Key Concepts

Polar Coordinates

Polar coordinates provide a way of locating a point in a plane using a distance and an angle. Instead of the typical x and y coordinates, polar coordinates \(r, \theta\) use:

• \(r\) - the distance from the origin (center point).
• \(\theta\) - the angle measured from the positive x-axis.

The positive direction of the angle is counter-clockwise. This system is particularly useful in scenarios involving circular or rotational symmetry, such as navigation or describing circular motions.

A key aspect to remember is that \(r\) can be positive or negative. A negative \(r\) essentially means you move in the opposite direction of the angle \(\theta\). Thus, understanding how these coordinates represent specific points is crucial for conversions.

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, are the most common way of pinpointing a location in a plane using two values, \(x\) and \(y\). These values represent:

• \(x\) - the horizontal distance from the origin.
• \(y\) - the vertical distance from the origin.

This system is widely used in graphing equations or functions, and it's straightforward due to its grid-like structure.

When converting from polar to rectangular coordinates, we use trigonometric functions to break down the radial distance \(r\) into \(x\) and \(y\). This transformation allows for easier operations like addition, subtraction, and visualization of complex plots.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are pivotal when converting between polar and rectangular coordinates. These functions relate the angle \(\theta\) and the radial distance \(r\) to the Cartesian coordinates:

• \(x = r \cdot \cos(\theta)\) - calculates the horizontal component.
• \(y = r \cdot \sin(\theta)\) - computes the vertical component.

Sine and cosine functions are derived from the unit circle, which is a circle with a radius of 1 centered at the origin. Each point on the unit circle has coordinates that correspond to the cosine and sine of the angle from the x-axis.

These relationships simplify understanding and computing values in physics, engineering, and computer graphics, where rotations and angles are commonly involved.