Question

For the following exercises, find the polar equation for the curve given as a Cartesian equation. $$ x+y=5 $$

Short answer

The polar equation is \(r = \frac{5}{\cos \theta + \sin \theta}\).

Step-by-step solution

Understand the Relationship Between Cartesian and Polar Coordinates

In Cartesian coordinates, points are represented as \((x, y)\), while in polar coordinates, points are represented as \((r, \theta)\). The transformations between these coordinate systems are given by the equations: \[ x = r \cos \theta \] \[ y = r \sin \theta \] \[ r = \sqrt{x^2 + y^2} \]\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

Substitute Cartesian Equations into Polar Equations

Starting from the given equation \(x + y = 5\), we substitute the polar transformations:\[ r \cos \theta + r \sin \theta = 5 \] by replacing \(x\) with \(r \cos \theta\) and \(y\) with \(r \sin \theta\).

Factor Out the Common Term

The equation \(r \cos \theta + r \sin \theta = 5\) can be simplified by factoring out \(r\):\[ r(\cos \theta + \sin \theta) = 5 \]

Solve for r

Solve the factored equation for \(r\):\[ r = \frac{5}{\cos \theta + \sin \theta} \]This gives the polar equation for the curve originally given in Cartesian form.

Key concepts

Cartesian Coordinates

Cartesian coordinates are a fundamental system used in mathematics to identify the position of a point in a plane. It uses two numerical values, known as coordinates, presented as \(x, y\). These values represent the horizontal and vertical distances of a point from a fixed reference point called the origin (0,0). This system is highly beneficial in geometry and algebra because of its intuitive grid-like structure.
For example, in the equation \(x + y = 5\), each solution pair \(x, y\) determines the position of a point on the graph. When graphed, these solutions form a straight line that shows all combinations of \(x, y\) adding up to 5, demonstrating the direct, linear relationship between these variables.

Coordinate Transformation

Coordinate transformation is the process of converting one set of coordinates into another, often from Cartesian to polar or vice-versa. This conversion is facilitated by mathematical formulas that relate the two coordinate systems.
For converting Cartesian coordinates \(x, y\) to polar coordinates \(r, \theta\), the following equations are used:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

These equations help us transition between the two systems seamlessly, allowing us to explore and solve geometric problems with increased flexibility and clarity.

Polar Equations

Polar equations express mathematical relationships using polar coordinates \(r, \theta\). In this system, \(r\) represents the distance from the origin, while \(\theta\) denotes the angle formed with the positive x-axis. Polar equations often simplify the representation of curves and circles, highlighting the radial symmetry that Cartesian coordinates may obscure.
In the conversion of \(x + y = 5\) to a polar equation, we use the relations \(x = r \cos \theta\) and \(y = r \sin \theta\), letting us derive the polar form \(r(\cos \theta + \sin \theta) = 5\). Solving it for \(r\), as \(r = \frac{5}{\cos \theta + \sin \theta}\), we get the curve's polar equation, showcasing the circle-like characteristics of the original line equation in a new light.

Curve Conversion

Curve conversion involves translating the depiction of a geometric shape or curve from one coordinate system to another. In practical mathematical problems, converting the representations of curves between Cartesian and polar forms can offer simpler solutions or provide new insights into their properties.
The curve defined by \(x + y = 5\), when transformed into its polar form \(r = \frac{5}{\cos \theta + \sin \theta}\), provides an alternative visualization. This transformation highlights how line equations can take different characteristic forms under varying systems. Through curve conversion, we can understand the underlying geometry of the curve more deeply, whether analyzing linearity in Cartesian coordinates or radial patterns in polar coordinates.