Question

Convert the polar equation to a rectangular equation. \(r=\tan \theta\)

Short answer

The rectangular form is \( x^4 + x^2y^2 - y^2 = 0 \).

Step-by-step solution

Express in Terms of Sine and Cosine

The equation given is in the polar form: \( r = \tan \theta \). Start by expressing \( \tan \theta \) in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This gives us the equation \( r = \frac{\sin \theta}{\cos \theta} \).

Remove the Fraction by Multiplying by \(\cos \theta\)

Multiply both sides of the equation by \( \cos \theta \) to eliminate the fraction. Doing so, we get \( r \cos \theta = \sin \theta \).

Utilize Polar to Rectangular Coordinate Conversion

Recall the conversion formulas from polar to rectangular coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute \( r \cos \theta = x \) and \( \sin \theta = \frac{y}{r} \) into the equation. This converts the equation to \( x = \frac{y}{r} \).

Express \( r \) in Rectangular Terms

Using the identity \( r = \sqrt{x^2 + y^2} \), substitute \( r \) in the rectangular terms: \( x = \frac{y}{\sqrt{x^2 + y^2}} \).

Eliminate the Fraction

Multiply both sides of the equation by \( \sqrt{x^2 + y^2} \) to eliminate the fraction: \( x \sqrt{x^2 + y^2} = y \).

Square Both Sides to Simplify

Square both sides of the equation to remove the square root, resulting in \( x^2 (x^2 + y^2) = y^2 \).

Simplify the Expression

Expand the equation \( x^2 (x^2 + y^2) = y^2 \) to \( x^4 + x^2y^2 = y^2 \).

Rearrange the Equation

Rearrange the equation to bring all terms involving \( x \) and \( y \) to one side: \( x^4 + x^2y^2 - y^2 = 0 \). This is the rectangular form of the given polar equation.

Key concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when dealing with equations involving angles. These identities allow us to express trigonometric functions in terms of sine and cosine. In our polar to rectangular conversion exercise, we use the identity for the tangent function:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This identity breaks down the tangent into its sine and cosine components, making it easier to manipulate algebraically.
We start with the polar equation \( r = \tan \theta \), and by substituting the identity, we convert this into \( r = \frac{\sin \theta}{\cos \theta} \). Understanding and applying these identities are fundamental steps in transitioning equations between different forms.

Coordinate Systems

Coordinate systems serve as the foundation for describing positions on a plane. Two common systems are the polar and rectangular (Cartesian) coordinate systems.
The polar coordinate system uses the variables \( r \) and \( \theta \), representing the radial distance and the angle from a reference direction, respectively. Conversely, the rectangular system uses \( x \) and \( y \) as coordinates along horizontal and vertical axes.

• Polar to rectangular conversion involves using the relationships:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)

In our exercise, these formulas help us convert from a polar equation to a rectangular form by substituting \( x \) for \( r \cos \theta \) and \( y \) for \( r \sin \theta \). Recognizing how to do these substitutions is key when switching between coordinate systems.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations. It plays a critical role when converting equations from one form to another, like from polar to rectangular coordinates.
Here, once we express \( r = \tan \theta \) in terms of \( x \) and \( y \), we face the task of algebraically manipulating the terms until we achieve a suitable form.

• Step-by-step, we:
  • Eliminate fractions by multiplying both sides by expressions like \( \sqrt{x^2 + y^2} \).
  • Square both sides to simplify and remove radicals.
  • Rearrange the resulting terms to bring all expressions involving \( x \) and \( y \) to the same side.

These techniques of manipulating expressions help us eventually arrive at the simplified rectangular equation \( x^4 + x^2y^2 - y^2 = 0 \). Mastery of these algebraic techniques is essential for handling complex conversions and simplifications in mathematics.