Problem 56 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=6 \cos \theta $$

Step-by-Step Solution

Answer

The rectangular form is \( x = 6 \).

Step 1: Understand the Polar Equation

The given polar equation is \( r = 6 \cos \theta \). In this equation, \( r \) represents the distance from the origin to a point, and \( \theta \) is the angle from the positive x-axis to the line connecting the origin to that point.

Step 2: Use the Conversion Formulas

To convert polar coordinates to rectangular coordinates, we use the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). These formulas express the rectangular coordinates \((x,y)\) in terms of \(r\) and \(\theta\).

Step 3: Replace \(r\cos\theta\) with \(x\)

From the polar equation \( r = 6 \cos \theta \), multiply both sides by \( r \): \( r^2 = 6r\cos\theta \). We know that \( r\cos\theta = x \). Substitute \( x \) in place of \( r\cos\theta \), yielding \( x = 6 \).

Step 4: Formulate the Rectangular Equation

Since we have expressed \( r^2 = 6x \) and \( y^2 = r^2 - x^2 \) and know from the polar form \( r^2 = x^2 + y^2 \), the substitution \( r = x \) provides our rectangular equation. The final rectangular form is \( x = 6 \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are crucial in mathematics, particularly in two-dimensional geometry. They consist of pairs \(x, y\), where \(x\) and \(y\) represent the respective horizontal and vertical distances from the origin. This system is familiar if you've ever plotted points on a grid or graph paper.

- **Coordinate Planes**: Points are placed on a plane with a horizontal \(x\)-axis and a vertical \(y\)-axis.
- **Quadrants**: The plane is divided into four quadrants which help identify the sign of \(x\) and \(y\).
- **Distance Representation**: \(x\) shows how far to move left or right from the origin, while \(y\) shows the up or down movement.

Understanding rectangular coordinates helps us visualize mathematical functions and shapes better. This coordinate system is widely used due to its simplicity and straightforward representation of algebraic equations.

Polar Equations

Polar equations offer a different approach to defining points in a plane. In a polar coordinate system, each point is defined by a distance and an angle, \(r\) and \(\theta\), respectively.

- **Distance from Origin**: \(r\) refers to how far a point is from the origin. If \(r\) is positive, the point lies directly along the direction of the angle \(\theta\).
- **Angle Measurement**: \(\theta\) is the angle formed with the positive \(x\)-axis, typically measured in radians.
- **Flexible System**: Polar coordinates can describe points more naturally in cases involving circular paths or rotations.

To convert a polar equation like \(r = 6 \cos \theta\) to rectangular form, we utilize relationships involving \(r\) and \(\theta\) to derive corresponding \(x, y\) coordinates. This transformation often involves leveraging trigonometric identities and relationships to simplify and express the equations in rectangular form.

Coordinate Transformation

Transforming from one coordinate system to another involves specific mathematical procedures. Conversion between polar and rectangular coordinates helps in analyzing and solving various geometric and real-world problems.

- **Conversion Formulas**: A crucial conversion involves the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\), directly relating polar values to their rectangular counterparts.
- **Rectangular Representation**: To express a polar equation in rectangular form, substitute polar-based expressions using these formulas. For instance, the polar equation \(r = 6 \cos \theta\) is converted by noting \(r \cos \theta = x\), ultimately leading to the rectangular equation \(x = 6\).
- **Solving Polar to Rectangular**: This involves substituting known polar values to solve for \(x\) and \(y\), and may require simplifying expressions using identities.

Understanding coordinate transformation allows you to switch between systems seamlessly, making it easier to choose the most suitable system for a given problem. This flexibility is extremely beneficial when dealing with complex geometrical shapes or paths.