Question

Write the rectangular coordinates of each point. $$\left(6.30,227^{\circ}\right)$$

Short answer

\(x = -4.45\), \(y = -4.45\sqrt{3}\)

Step-by-step solution

Understand Polar Coordinates

Polar coordinates are given in the form \(r, \theta\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured from the positive x-axis. The given polar coordinates are \(6.30, 227^\circ\).

Convert Angle to Radians

To convert degrees to radians, use the conversion factor \(\pi \text{ radians} = 180^\circ\). Thus, \(227^\circ = \frac{227\pi}{180} \text{ radians}\).

Calculate Rectangular Coordinates

Use the polar to rectangular coordinate conversion formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Here, \(r = 6.30\) and \(\theta = \frac{227\pi}{180}\). Calculate \(x = 6.30 \cos(\frac{227\pi}{180})\) and \(y = 6.30 \sin(\frac{227\pi}{180})\).

Simplify to Find x and y

Upon calculation, \(x\) and \(y\) can be found using a calculator to compute the cosine and sine values of \(\theta\). Round if necessary for practical purposes.

Key concepts

Polar Coordinates

Polar coordinates are an alternative to the more familiar rectangular (or Cartesian) coordinate system for representing points in a plane. Unlike rectangular coordinates, which use a grid of horizontal and vertical lines, polar coordinates measure the location of a point based on its distance from a central point (called the origin or pole) and the angle from a reference direction, typically the positive x-axis.

In polar coordinates, a point is denoted as \(r, \theta\), where \(r\) is the radius or radial distance and \(\theta\) is the angular component known as the argument or polar angle. Here's how it works:

• The radial distance (\(r\)) is the straight-line distance from the origin to the point.
• The angle (\(\theta\)) is measured in degrees or radians and indicates the direction of \(r\) with respect to the positive x-axis.

Understanding these definitions is the first step in converting from polar to rectangular coordinates.

Rectangular Coordinates

Rectangular (or Cartesian) coordinates are the most common coordinate system used in mathematics and science, built around two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Any point in this two-dimensional space can be identified with a unique pair of numbers, \( (x, y) \), which represent the horizontal and vertical distances from the origin.

Here's what's important to remember about rectangular coordinates:

• The x-coordinate indicates how far to move left or right from the origin.
• The y-coordinate shows how far to move up or down from the origin.
• This system is what you see on most graphs and grid paper used for plotting functions.

When we talk about converting polar coordinates to rectangular coordinates, we’re essentially translating radial distances and angles into these horizontal and vertical distances.

Angle Conversion

Angle conversion is a necessary skill when working with polar coordinates since angles can be expressed in degrees or radians. Radians are often preferred in higher mathematics and the sciences because they provide a direct link between the angle and the arc length on a circle.

To convert degrees to radians, we use the conversion factor \(\pi \text{ radians} = 180^\circ\). Conversely, to convert from radians to degrees, we multiply by \(\frac{180^\circ}{\pi}\). Here is a simple way to remember it:

• To go from degrees to radians, multiply by \(\frac{\pi}{180^\circ}\).
• To go from radians to degrees, multiply by \(\frac{180^\circ}{\pi}\).

The ability to convert between these two units is essential when we want to use trigonometric functions to find rectangular coordinates, as many calculators and programming languages require angles to be in radians.

Trigonometric Functions

Trigonometric functions are a cornerstone of geometry and are fundamentally related to circles and angles. They relate the angles of a triangle to the lengths of its sides. In the context of polar to rectangular coordinate conversion, we mainly use the sine (\(\sin\)) and cosine (\(\cos\)) functions.

Here’s what you need to know about these functions:

• \(\cos(\theta)\) gives the horizontal coordinate of a unit circle at an angle \(\theta\).
• \(\sin(\theta)\) gives the vertical coordinate of a unit circle at an angle \(\theta\).
• They can be used to find the x and y coordinates in the rectangular system by multiplying the radial distance \(r\) by \(\cos(\theta)\) and \(\sin(\theta)\), respectively.

By applying these functions to the polar angle (\(\theta\)) and radial distance (\(r\)), we can accurately find the equivalent point in rectangular coordinates.