Problem 63 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=\frac{4}{1+2 \sin \theta} $$

Step-by-Step Solution

Answer

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

Step 1: Substitute Polar Coordinates

The given polar equation is \( r = \frac{4}{1 + 2\sin\theta} \). Recall the polar-to-rectangular transformations: \( r = \sqrt{x^2 + y^2} \), \( x = r \cos\theta \), and \( y = r \sin\theta \). We'll use these to convert the equation.

Step 2: Solve for Sin(θ)

From the transformations, \( \sin\theta = \frac{y}{r} \). Substitute this into the polar equation: \[ r = \frac{4}{1 + 2\left(\frac{y}{r}\right)} \] which simplifies to \[ r = \frac{4r}{r + 2y} \].

Step 3: Eliminate Fractions

Multiply both sides of the equation by \( r + 2y \) to eliminate the fraction: \[ r(r + 2y) = 4r \].

Step 4: Expand and Rearrange

Distribute on the left: \[ r^2 + 2ry = 4r \]. Now, solve for terms involving \( r \): \[ r^2 + 2ry - 4r = 0 \].

Step 5: Substitute Rectangular Coordinates

Recognize that \( r = \sqrt{x^2 + y^2} \). Substitute back into the equation: \[ (\sqrt{x^2 + y^2})^2 + 2y\sqrt{x^2 + y^2} - 4\sqrt{x^2 + y^2} = 0 \] which simplifies to \[ x^2 + y^2 + 2y\sqrt{x^2 + y^2} = 4\sqrt{x^2 + y^2} \].

Step 6: Simplify Further

Factor out \( \sqrt{x^2 + y^2} \) from the terms where possible: \[ (\sqrt{x^2 + y^2})(x^2 + y^2 - 4 + 2y) = 0 \]. Simplify to \[ x^2 + (y-2)^2 = 4 \].

Step 7: Identify the Resulting Equation

We have determined that \( x^2 + (y-2)^2 = 4 \), which is the rectangular form of the given polar equation. This represents a circle with center \( (0, 2) \) and radius 2.

Key Concepts

Polar Coordinates

Polar coordinates are a way to define a point in a two-dimensional space using a radius and an angle. Imagine a circle drawn on a piece of paper where a point can be defined by how far it is from the center (radius, \( r \)) and how many degrees or radians around the circle it is from the starting line (angle, \( \theta \)).
To visualize it, think of a compass pointing north. If the compass needle is 60 degrees to the right and 4 units away from the center, in polar coordinates that point is described as \((r, \theta) = (4, 60^\circ)\).

• \( r \) is the distance from the origin to the point.
• \( \theta \) is the angle measured from the positive x-axis.

In mathematics, angles are often measured in radians. Polar coordinates are particularly useful in scenarios where relationships depend on distances from a central point and angles, like the path traced by a rotating object.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent a point in space using two numbers: \( x \) and \( y \). Here, each coordinate corresponds to a unique location by indicating the distance from two perpendicular reference lines, typically referred to as the x-axis and the y-axis.
Think of a piece of graph paper where each square represents one unit. A point (3, 2) means it is 3 units to the right of the origin and 2 units up. This system is widely used because it directly translates to our natural perception of up, down, left, and right movements.

• \( x \) is the horizontal position, positive to the right.
• \( y \) is the vertical position, positive upward.

Rectangular coordinates are ideal for equations where relationships are modeled as direct linear distances along these axes.

Equation Conversion

Converting equations between polar and rectangular forms helps in different analytical interpretations and solutions to problems. The conversion relies on geometric relationships between polar and rectangular coordinates.
Key formulas often used in this conversion are:

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

For example, converting the polar equation \( r = \frac{4}{1 + 2\sin\theta} \) into rectangular coordinates involves substituting polar parameters with their rectangular equivalents to derive a more familiar form like \( x^2 + (y-2)^2 = 4 \), which represents a circle. This reveals the shape geometrically represented by the equation.

Circle Equation

In rectangular coordinates, a circle's equation typically looks like \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \((h, k)\) is the circle's center, and \( r \) is the radius. This formula helps you easily find where a circle is on the graph and how big it is.
Rewriting the polar to rectangular conversion shows a circle centered at specific coordinates with a given radius. For example, the conversion resulting in \( x^2 + (y - 2)^2 = 4 \) reveals a circle with:

• Center: \( (0, 2) \)
• Radius: \( 2 \)

Understanding circles in this form visualizes not only the circle's shape but also its position in space, which is essential for geometric interpretations in many fields like engineering and physics.