Problem 54 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall Conversion Formulas

To convert polar coordinates to rectangular coordinates, we use the following equations: \( x = r \cos \theta \) and \( y = r \sin \theta \). Additionally, we can use \( r^2 = x^2 + y^2 \) and \( \tan \theta = \frac{y}{x} \).

Step 2: Substitute \( r \) in Terms of \( y \)

Start by expressing \( r \) in the polar form in terms of \( \sin \theta = \frac{y}{r} \). The given equation is \( r = \frac{1}{1 + \sin \theta} \). Substitute \( \sin \theta = \frac{y}{r} \) to get \( r = \frac{1}{1 + \frac{y}{r}} \).

Step 3: Simplify the Expression

Multiply both sides of the equation by \( 1 + \frac{y}{r} \) to eliminate the fraction: \[ r(1 + \frac{y}{r}) = 1 \] This simplifies to: \[ r + y = r \] which further simplifies to: \[ r - y = 1 \].

Step 4: Express \( r \, (x, y) \) in Rectangular Form

Recognizing any possible mishap from simplification, check the simplification again or try alternates: Substitute back for \( r \) using \( r = \frac{1}{1 + \sin \theta} \), get, from the initial variable definition, i.e., \( r = \frac{1}{1 + \frac{y}{r}} \)Rewriting multiplication from previous clearings, \[ r^2 = r - y \] leads to substitute expressions using \( r^2 = x^2+y^2\), hence derive:\[ x^2 + y^2 - r = y = 1 \].

Step 5: Verify and Solve

Rearranging the simplified equation:Utilize multiplication, \( x^2 + (y - 1)^2 = 1 \) to verify that conversions and expressions form a circle,Hence this itself gives out ideal rectangular coordinates meaning using \( x,x_0 and x,x \).

Key Concepts

Polar to Rectangular Conversion

The process of converting polar coordinates to rectangular coordinates involves using key relationships between the systems. In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. To convert this to rectangular coordinates, we use the formulas:

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

These formulas are derived by understanding that \(r\) serves as the hypotenuse of a right triangle, with \(x\) and \(y\) as the other two sides. By using trigonometric identities, specifically \(\cos \theta\) and \(\sin \theta\), we can express \(x\) and \(y\) in terms of \(r\) and \(\theta\).
This transformation allows us to describe the same point in the Cartesian coordinate system, which is helpful in various mathematical contexts including graphing functions or solving equations geometrically.

Polar Coordinates

Polar coordinates offer a unique way of representing points in a plane, focusing on the distance and angle relative to the origin. Unlike rectangular coordinates which depend on x and y values, polar coordinates describe positions through:

• The radial coordinate \(r\),
• And the angular coordinate \(\theta\).

The radial coordinate \(r\) indicates how far the point is from the origin, while the angle \(\theta\) specifies the direction of the point from the positive x-axis. This system is particularly useful in scenarios involving circular and spiral patterns.
For example, when dealing with rotations or trajectories that pivot around a central point, polar coordinates often make complex problems more manageable. However, to apply these in rectangular frameworks, conversion methods are employed, thus bridging geometrical interpretations across systems.

Coordinate Geometry

Coordinate geometry, or Cartesian geometry, is a fundamental aspect of understanding spatial relationships and patterns through coordinates. This branch of mathematics combines algebra and geometry, using a plane with an x-axis and a y-axis to define the position of points. The power of coordinate geometry lies in its ability to simplify problems involving shapes and trajectories.

• Equations of lines, curves, and circles can be easily understood using this framework.
• The conversion between polar and rectangular coordinates exemplifies problem-solving across different coordinate types.

For instance, a circle can be described geometrically with polar coordinates and translated to rectangular form, offering another way of tackling complex geometric problems.
This dual perspective enriches our approach to mathematical equations, enhancing both our understanding and problem-solving abilities. In essence, coordinate geometry acts as a unifying language of mathematics, enabling seamless transitions and interpretations between different systems.