Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(5, \frac{5 \pi}{3}\right) $$

Short answer

The rectangular coordinates are (2.5, -2.5\sqrt{3}).

Step-by-step solution

- Understand the Problem

The polar coordinates \(\theta, \ r\) can be converted to rectangular coordinates \(x, y\) using the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). In this case, \(r = 5\) and \(\theta = \frac{5 \pi}{3}\).

- Calculate x-coordinate

Using the formula \(x = r \cos(\theta)\), calculate \(x\): \ \[ x = 5 \cos\left(\frac{5\pi}{3}\right) \] \ Since \(\frac{5\pi}{3} \) is in the fourth quadrant, \(\cos(\frac{5\pi}{3}) = \cos\left(2\pi - \frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\), thus \[ x = 5 \left(\frac{1}{2}\right) = 2.5 \]

- Calculate y-coordinate

Using the formula \(y = r \sin(\theta)\), calculate \(y\): \ \[ y = 5 \sin\left(\frac{5\pi}{3}\right) \] \ Since \(\frac{5\pi}{3}\) is in the fourth quadrant, \(\sin\left(\frac{5\pi}{3}\right) = \sin\left(2\pi - \frac{\pi}{3}\right) \) and \(\sin(2\pi - k) = -\sin(k)\), thus \[ y = 5 \left(-\sin\left(\frac{\pi}{3}\right)\right) = 5 \left(-\frac{\sqrt{3}}{2}\right) = -2.5\sqrt{3} \]

- Combine x and y Coordinates

The rectangular coordinates are \(x, y\). Therefore, \ \(\left(5, \frac{5\pi}{3}\right) = (2.5, -2.5\frac{\sqrt{3}})\).

Key concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system that allows us to specify each point uniquely in a plane by a pair of numerical coordinates. These coordinates are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The two coordinates are usually labeled as \( (x, y) \).
The x-coordinate represents the horizontal distance, while the y-coordinate represents the vertical distance.
This coordinate system is widely used in algebra, geometry, and various applied sciences because it is straightforward and intuitive. When working on a problem that requires converting from polar to rectangular coordinates, understanding the basic components of the Cartesian plane is essential.
Here are key points to remember:

• The x-axis runs horizontally, while the y-axis runs vertically.
• A point on the plane is defined by how far along the x-axis and y-axis it is.
• Quadrants indicate in which part of the plane the point is located. Each quadrant has unique sign conventions for x and y.

Being comfortable with rectangular coordinates helps in visualizing problems more easily and solving them effectively.

polar coordinates

Polar coordinates are another way of representing points in a plane, mainly used in fields where the relationship to a central point (origin) is more meaningful. This system represents points not by linear distances, but by an angle and a radius. The general form of polar coordinates is \( (r, \theta) \).

• The radial coordinate \(r\): This represents the distance from the origin to the point.
• The angular coordinate \(\theta\): This is the angle measured from the positive x-axis to the line connecting the origin with the point.

Understanding these two components makes it easier to make conversions and interpretations.
For instance, in the given problem, converting \( (5, \frac{5\pi}{3})\) to rectangular coordinates involves interpreting \(r - 5\) as the radial distance.
The angle \(\theta\) is translated from its position relative to the positive x-axis. It might be helpful to visualize it on a graph, understanding which quadrant it lies in, and how trigonometric functions, like sine and cosine, apply to these angles.

trigonometric functions

Trigonometric functions are crucial in the conversion from polar to rectangular coordinates. The key functions involved are cosine and sine. These functions help us break down the radial distance \(r\) into its horizontal \(x\) and vertical \(y\) components.
Here’s a brief overview:

• Cosine function: \(\cos(\theta)\) gives the ratio of the adjacent side to the hypotenuse in a right triangle.
• Sine function: \(\sin(\theta)\) gives the ratio of the opposite side to the hypotenuse.

The formulas \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\) are derived from these definitions.
For example, in the provided exercise, to find the x-coordinate and y-coordinate, we use:
<\section> For \(x\): \[ x = 5 \cos\left(\frac{5\pi}{3}\right) = 2.5 \]
<\section> For \(y\): \[ y = 5 \sin\left(\frac{5\pi}{3}\right) = -2.5\frac{\sqrt{3}}{2} \]
The principles behind these calculations provide a firm foundation for tackling numerous mathematical problems involving different coordinate systems.