Problem 62 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + y^2 - y = 1 \).

Step 1: Understand the Relationships

To convert from polar to rectangular coordinates, recall the relationships: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). Also note that \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \).

Step 2: Substitute into the Equation

Substitute \( \sin \theta = \frac{y}{r} \) into the given polar equation. This gives: \( r = \frac{1}{1 + \frac{y}{r}} \).

Step 3: Simplify the Equation

Multiply both sides of the equation by \( r(1 + \frac{y}{r}) \) to eliminate the fraction. The equation becomes: \( r^2 = 1 + y \).

Step 4: Substitute for \( r^2 \)

Replace \( r^2 \) with \( x^2 + y^2 \). So the equation becomes: \( x^2 + y^2 = 1 + y \).

Step 5: Simplify Further

Rearrange the equation to make it look neater. Subtract \( y \) from both sides: \( x^2 + y^2 - y = 1 \).

Key Concepts

Rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system of defining points in a plane using two numbers:

• \( x \): the horizontal distance from the origin
• \( y \): the vertical distance from the origin

This coordinate system is based on perpendicular axes, often labeled as the \( x \)-axis and \( y \)-axis.
Each point can be expressed as \( (x, y) \). In mathematics and physics, rectangular coordinates are frequently used due to their simplicity for defining locations, describing graphs of functions, and solving equations.
Understanding these coordinates is key when converting other forms of representation, such as polar coordinates, into a format that is often more intuitive and easier to work with.

Polar coordinates

Polar coordinates offer a different way of representing points in a plane. Instead of describing a point with horizontal and vertical distances, polar coordinates are defined using:

• \( r \): the radial distance from a fixed point known as the origin
• \( \theta \): the angle from a fixed direction, usually the positive \( x \)-axis

A point in polar coordinates is written as \( (r, \theta) \).
This method is particularly useful when dealing with circular or spiral shapes, where the relationship with the origin and angles is more apparent than right-angle distances.
Learning to express points in polar coordinates can greatly simplify problems where symmetry and curves are important.

Coordinate conversion

Converting between polar and rectangular coordinates involves a set of relationships. These relationships stem from trigonometry and are crucial for moving from one system to another. For polar to rectangular conversion:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These equations allow us to translate a point expressed in polar coordinates, \( (r, \theta) \), to its rectangular form, \( (x, y) \).
The reverse is also possible by using:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \arctan\left(\frac{y}{x}\right) \)

Mastering coordinate conversion is essential for tackling problems in different mathematical contexts and recognizing patterns in both coordinate systems.

Trigonometric relationships

Trigonometric relationships are the links that tie angles to side lengths in right triangles. These relationships are foundational in the conversion between polar and rectangular coordinates. Key identities include:

• \( \, \sin \theta = \frac{y}{r} \)
• \( \, \cos \theta = \frac{x}{r} \)
• \( \, \tan \theta = \frac{y}{x} \)

These concepts allow us to express complex equations and solve for unknowns effortlessly.
Understanding these trigonometric principles simplifies coordinate conversions, allowing easier manipulation and understanding of geometrical shapes.
When working with equations, knowing how to use these relationships to express terms in different forms is crucial. This builds a solid foundation for further exploration and application in various areas of mathematics and science.