Problem 24 - Algebra and Trigonometry

Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r=2 \sin \theta $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + (y - 1)^2 = 1 \). It represents a circle with center \( (0, 1) \) and radius \( 1 \).

Step 1: Write the given polar equation

The given polar equation is \[r = 2 \sin \( \theta \).\]

Step 2: Use the polar to rectangular conversions

Recall the polar to rectangular coordinate conversions: \[x = r \cos \( \theta \), y = r \sin \( \theta \), \cos \( \theta \) = \frac\{ x \} \{ r \}, and \sin \( \theta \) = \frac\{ y \} \{ r \}.\]

Step 3: Substitute \( \sin \( \theta \) = \frac\{ y \} \{ r \} \)

Substitute \( \sin \( \theta \) = \frac\{ y \} \{ r \} \) into the polar equation \( r \sin \( \theta \) = 2 \):\[ r = 2 \sin \( \theta \) = 2 \frac\{ y \} \{ r \} \ \rightarrow r^2 = 2y. \]

Step 4: Use the relationship between \(r^2 \) and \( x^2 \) and \(y^2\)

Recall that \( r^2 = x^2 + y^2 \), so substituting gives:\[ x^2 + y^2 = 2y. \]

Step 5: Convert to standard form of a conic

Rewrite the equation to complete the square for \( y \) terms:\[ x^2 + y^2 - 2y = 0 \ \rightarrow x^2 + (y^2 - 2y + 1) = 1 \ \rightarrow x^2 + (y - 1)^2 = 1. \]

Step 6: Identify the equation

This is the equation of a circle with center at \( (0, 1) \) and radius \( 1 \).

Step 7: Graph the equation

Graph a circle centered at \( (0, 1) \) with a radius of \( 1 \).

Key Concepts

Polar Equations

Polar equations use the polar coordinate system, represented by radius (\(r\)) and angle (\(\theta\)). These equations highlight relationships between these two quantities. Rather than specifying a point's exact location using \(x\) and \(y\) coordinates, polar equations describe where the point is based on how far it is from the origin (radius) and the angle from the positive x-axis. Common polar equations include forms like \(r = 2 + 3\theta\) or \(r = 4\sin(\theta)\). These equations can describe many shapes, such as circles, spirals, and other conic sections.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, use an \(x\) and \(y\) plane to define the position of points. Each point is represented by an ordered pair \((x, y)\), indicating distances from the x-axis and y-axis. This system is useful for representing linear and simple quadratic equations. For instance, the equation of a circle can be represented in this form as \(x^2 + y^2 = r^2\). Transforming polar coordinates to rectangular coordinates can encapsulate complex shapes in a simpler, more familiar coordinate system.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. Examples include circles, ellipses, parabolas, and hyperbolas. The equations of these curves can be expressed in both polar and rectangular forms. Circle equations, for example, turn into simple quadratic forms like \(x^2 + y^2 = r^2\), while ellipses and parabolas have more complex representations. Converting polar equations into rectangular form often reveals the conic section being described.

Circle Equation

A circle's equation in rectangular coordinates looks like \(x^2 + y^2 = R^2\), where \((x, y)\) are coordinates of points on the circle, and \(R\) is the radius. The equation can also be shifted to accommodate circles not centered at the origin, leading to forms like \((x-h)^2 + (y-k)^2 = R^2\), where \((h, k)\) is the circle's center. In the solution provided:

• We started with a polar equation \(r = 2\sin(\theta)\).
• Converted it to rectangular form \(x^2 + (y - 1)^2 = 1\).
• Identified that this represents a circle with center \( (0,1)\) and radius 1.

Coordinate Conversions

Converting between polar and rectangular coordinates is a key mathematical skill. The conversions are:

• From polar to rectangular: \(x = r\cos(\theta)\), \(y = r\sin(\theta)\).
• From rectangular to polar: \(r = \sqrt{x^2 + y^2}\), \(\theta = \arctan(\frac{y}{x})\).

In our example, we achieved this by substituting \(\sin(\theta) = \frac{y}{r}\) and \(r^2 = x^2 + y^2\) into the original polar equation. This allowed us to express a polar circle equation in rectangular form, making it easier to graph and understand its shape.