Problem 21 - Precalculus

Question

Convert each point to exact rectangular coordinates. $$\left(4, \frac{5 \pi}{3}\right)$$

Step-by-Step Solution

Answer

The rectangular coordinates are \((2, -2\sqrt{3})\).

Step 1: Identify Polar Coordinates

The polar coordinates given are \((r, \theta) = \left(4, \frac{5\pi}{3}\right)\), where \(r\) is the radius and \(\theta\) is the angle in radians.

Step 2: Compute the x-coordinate

Use the formula for the x-coordinate in rectangular form, \(x = r \cdot \cos(\theta)\). Substitute the given values: \(x = 4 \cdot \cos\left(\frac{5\pi}{3}\right)\). Since \(\cos\left(\frac{5\pi}{3}\right)\) is equivalent to \(\cos\left(\frac{-\pi}{3}\right)\) due to periodicity, and this equals \(\frac{1}{2}\), hence \(x = 4 \cdot \frac{1}{2} = 2\).

Step 3: Compute the y-coordinate

Use the formula for the y-coordinate in rectangular form, \(y = r \cdot \sin(\theta)\). Substitute the given values: \(y = 4 \cdot \sin\left(\frac{5\pi}{3}\right)\). Since \(\sin\left(\frac{5\pi}{3}\right)\) is equivalent to \(\sin\left(\frac{-\pi}{3}\right)\) and this equals \(-\frac{\sqrt{3}}{2}\), thus, \(y = 4 \cdot -\frac{\sqrt{3}}{2} = -2\sqrt{3}\).

Step 4: Combine x and y Coordinates

The rectangular coordinates corresponding to the polar coordinates \(\left(4, \frac{5\pi}{3}\right)\) are given by \((x, y) = (2, -2\sqrt{3})\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of using the familiar x and y axes of the rectangular coordinate system, in polar coordinates we have two values:

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

The use of polar coordinates is especially common in situations where we are dealing with circular or rotational symmetry.
The angle \(\theta\) is usually measured in radians, where \[2\pi\text{ radians} = 360^\circ\].For the given polar coordinates \((4, \frac{5\pi}{3})\), the radius \(r = 4\) and the angle \(\theta = \frac{5\pi}{3}\). This means the point is 4 units away from the origin at an angle of \(\frac{5\pi}{3}\) radians. Such coordinates are quite useful in fields such as physics and engineering due to their ability to simplify the representation of complex problems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system for representing points on a plane using two perpendicular axes: the x-axis and the y-axis. In this system, a point is identified by an ordered pair \((x, y)\) representing its horizontal and vertical distances from the origin.
To convert from polar coordinates \((r, \theta)\) to rectangular coordinates, you need to calculate:

• The x-coordinate using the formula \(x = r \cdot \cos(\theta)\).
• The y-coordinate using the formula \(y = r \cdot \sin(\theta)\).

This conversion leverages trigonometry to transform a point from a radial system into a more linear or grid-like system.
For the example \((4, \frac{5\pi}{3})\):

• The x-coordinate is calculated as \(x = 4 \cdot \frac{1}{2} = 2\).
• The y-coordinate is calculated as \(y = 4 \cdot -\frac{\sqrt{3}}{2} = -2\sqrt{3}\).

Thus, the point \((4, \frac{5\pi}{3})\) in polar coordinates converts to \((2, -2\sqrt{3})\) in rectangular coordinates.

Trigonometric Functions

Trigonometric functions play a crucial role in the conversion between polar and rectangular coordinates. The primary trigonometric functions used in this process are cosine and sine.

• Cosine (\(\cos\)): Used to calculate the horizontal distance, or the x-coordinate, from the polar origin to the point. For an angle \(\theta\), \(\cos(\theta)\) determines how far along the x-axis a point will be relative to the unit circle.
• Sine (\(\sin\)): Used to calculate the vertical distance, or the y-coordinate, from the polar origin to the point. \(\sin(\theta)\) helps to determine the point's position along the y-axis.

When given the angle \(\theta = \frac{5\pi}{3}\), these functions are used to find the respective x and y components:

• \(\cos\left(\frac{5\pi}{3}\right)\) simplifies to \(\cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}\), giving us the x-coordinate.
• \(\sin\left(\frac{5\pi}{3}\right)\) simplifies to \(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\), providing the y-coordinate.

Understanding how to use these trigonometric functions with the unit circle is pivotal when performing conversions from polar to rectangular coordinates.