Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r=4 $$

Short answer

\( x^2 + y^2 = 16 \) represents a circle centered at the origin with radius 4.

Step-by-step solution

Understand Polar Coordinates

In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle from the positive x-axis.

Identify Given Polar Equation

The given polar equation is \(r=4\). This means that all points are at a distance of 4 units from the origin, regardless of the angle \(\theta\).

Express in Rectangular Coordinates

Use the conversion formulas from polar to rectangular coordinates. These formulas are \(x = r \cos \theta\) and \(y = r \sin \theta\). Since \(r=4\), we substitute \(r\) with the value to get: \(x = 4 \cos \theta\) and \(y = 4 \sin \theta\).

Eliminate the Angle \(\theta\)

Since \(x = 4 \cos \theta\) and \(y = 4 \sin \theta\), use the relationship \(x^2 + y^2 = r^2\) and substitute \(r = 4\). Thus, we get: \(x^2 + y^2 = 4^2 = 16\).

Identify and Graph the Equation

The equation \(x^2 + y^2 = 16\) represents a circle centered at the origin (0, 0) with a radius of 4. To graph it, draw a circle with center at the origin and radius extending 4 units in all directions.

Key concepts

polar coordinates

Polar coordinates are an alternative way to represent points in a plane. Instead of using Cartesian coordinates \((x, y)\), we use \((r, \theta)\). Here, \((r)\) represents the radial distance from the origin, and \((\theta)\) indicates the angle between the positive x-axis and the line connecting the point to the origin.

For example, the point (4, 45°) in polar coordinates translates to a point 4 units away from the origin at an angle of 45 degrees from the positive x-axis. This system is particularly useful in scenarios involving circular or rotational symmetry.

rectangular coordinates

Rectangular coordinates, or Cartesian coordinates, represent points in a plane using two values: \((x)\) and \((y)\). The \((x)\) value corresponds to the horizontal distance from the origin, and the \((y)\) value corresponds to the vertical distance.

In the equation \((x, y)\), both values can be positive or negative, depending on the point's location relative to the axes. For instance, the point (3, 4) is 3 units to the right of the origin and 4 units up. Conversely, (-3, -4) is 3 units left and 4 units down from the origin.

coordinate transformation

Coordinate transformation allows us to convert between polar and rectangular coordinates. This transformation is essential for solving problems that are easier in one system than another.

To convert from polar to rectangular coordinates:
1. Use \((x = r \cos \theta)\).
2. Use \((y = r \sin \theta)\).

Similarly, to convert from rectangular to polar coordinates:
1. Calculate the distance using \((r = \sqrt{x^2 + y^2})\).
2. Find the angle using \((\theta = \tan^{-1}(\frac{y}{x}))\).

Understanding these transformations can simplify complex problems and reveal relationships that may not be readily apparent in one coordinate system.

graphing equations

Graphing equations involves plotting points that satisfy a given equation on a coordinate system. When given an equation, it's crucial to know which coordinate system you're working in to properly graph it.

In polar coordinates, graphing involves plotting points based on their \((r)\) values and angles \((\theta)\). For example, the equation \((r = 4)\) represents all points 4 units away from the origin, forming a circle.

When converted to rectangular coordinates, the same equation becomes \((x^2 + y^2 = 16)\), which is the equation of a circle with a radius of 4 centered at the origin. Graphing this involves drawing a circle with its center at the origin and a radius that extends 4 units in all directions.