Problem 59 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=1+2 \sin \theta $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^4 + 2x^2y^2 + y^4 - 4x^2y - 4y^3 + 3y^2 - x^2 = 0 \).

Step 1: Recall Polar to Rectangular Formulas

To convert the given polar equation to rectangular coordinates, we use the following conversion formulas: \[ x = r \cos \theta \]\[ y = r \sin \theta \]\[ r^2 = x^2 + y^2 \] These relationships will help to rewrite the equation in terms of \(x\) and \(y\).

Step 2: Express r in Terms of y

In the given equation \( r = 1 + 2 \sin \theta \), substitute the formula \( y = r \sin \theta \). Rearrange it as \( \sin \theta = \frac{y}{r} \). Substitute it into the equation:\[ r = 1 + 2 \left(\frac{y}{r}\right) \]

Step 3: Clear the Fraction

Multiply both sides of the equation by \( r \) to clear the fraction:\[ r^2 = r + 2y \]

Step 4: Substitute r² in Terms of x and y

Use the formula \( r^2 = x^2 + y^2 \) to replace \( r^2 \) in the equation:\[ x^2 + y^2 = r + 2y \]

Step 5: Substitute r in Terms of x and y

Recognize that \( r = \sqrt{x^2 + y^2} \). Substitute this in place of \( r \) in the equation:\[ x^2 + y^2 = \sqrt{x^2 + y^2} + 2y \]

Step 6: Isolate the Square Root

Rearrange the equation to isolate the square root:\[ x^2 + y^2 - 2y = \sqrt{x^2 + y^2} \]

Step 7: Square Both Sides

Square both sides to eliminate the square root:\[ \left(x^2 + y^2 - 2y\right)^2 = x^2 + y^2 \]

Step 8: Expand and Simplify

Expand the left side and simplify:\[ x^4 + 2x^2y^2 + y^4 - 4x^2y - 4y^3 + 4y^2 = x^2 + y^2 \]

Step 9: Collect Like Terms

Rearrange the terms to get everything on one side of the equation:\[ x^4 + 2x^2y^2 + y^4 - 4x^2y - 4y^3 + 3y^2 - x^2 = 0 \]This is the equation in rectangular form.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using a distance and an angle. This method is particularly useful when dealing with problems involving circles and curves. Rather than the typical x and y axis, polar coordinates use:

• r: the length from the origin to the point.
• θ (theta): the angle from the positive x-axis to the line connecting the origin and the point.

This system can simplify the representation and calculations of points that are circularly symmetric or have radial patterns. Polar coordinates can be very powerful when the scenario or shape fits this circular logic. For example, spirals, circles, and waves work well with polar coordinates.

Rectangular Coordinates

Rectangular coordinates, sometimes referred to as Cartesian coordinates, involve the standard grid system designated by x and y values. These coordinates are straightforward for representing the position of a point on a simple plane. In a rectangular coordinate system:

• x is the horizontal distance from the origin.
• y is the vertical distance from the origin.

This system forms a grid and is ideal for plotting linear relationships and straightforward geometric figures like lines and rectangles. Crucially, rectangular coordinates are extremely useful for algebraic computations owing to their simple, straightforward layout. Transitioning from polar to rectangular involves combining circular motion (r and θ) into this linear grid format.

Coordinate Conversion

Coordinate conversion is the process of transforming one type of coordinate system into another. When converting from polar to rectangular coordinates, as outlined in the original exercise, a set of formulas is employed:

• Formula for x: \( x = r \cos \theta \)
• Formula for y: \( y = r \sin \theta \)
• Relation for r: \( r^2 = x^2 + y^2 \)

These conversion formulas enable the translation of a point or equation from the polar to the Cartesian plane. The given polar equation \( r = 1 + 2 \sin \theta \) transforms into a complex equation in x and y by working through these mathematical relationships. This conversion is significant when the problem requires visualization or manipulation within the standard grid system.

Trigonometric Functions

Trigonometric functions are mathematical relations based on angles and their standard position on a unit circle. In the context of coordinate conversion, these functions help link polar and rectangular coordinates. The primary trigonometric functions involved are:

• Sine (\( \sin \theta \)): represents the y-coordinate on the unit circle.
• Cosine (\( \cos \theta \)): represents the x-coordinate on the unit circle.

These functions are foundational as they allow us to relate the angle \( \theta \) in polar coordinates to the x and y coordinates in rectangular form. In our exercise, \( \sin \theta \) transforms the polar equation into an expression involving y, illustrating how trigonometric identities facilitate the conversion process. Understanding these functions is crucial as they underpin countless physics, engineering, and mathematics applications, connecting circular and linear systems.