Problem 100 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r^2=2\ \sin\ \theta\)

Step-by-Step Solution

Answer

The rectangular form of the equation \(r^2=2\ \sin\ \theta\) is \(x^2 + y^2 = 2y\).

Step 1: Identify the Given Polar Equation

The exercise provides the polar equation \(r^2=2\ \sin\ \theta\). The goal is to convert this polar equation into its equivalent Cartesian(rectangular) form.

Step 2: Use the relationship

We know from the polar to rectangular conversion relationships that \(y = r\ \sin\ \theta\) and \(r^2 = x^2 + y^2\). To transform the given polar equation into rectangular form, we can substitute \(y\) for \(r\ \sin\ \theta\) giving us \(r^2 = 2y\).

Step 3: Substitute r^2

Next we substitute \(r^2\) for \(x^2 + y^2\) giving us the equation in its rectangular form: \(x^2 + y^2 = 2y\).

Key Concepts

Polar Equation

A polar equation represents a relationship between the radial distance, 'r', and the angle, 'θ', in polar coordinates. These coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

In the given exercise, the polar equation is expressed as \(r^2=2\sin\theta\). This indicates that the value of \(r^2\), which is the square of the distance from the origin, varies as a function of the sine of the angle \(\theta\). Polar equations like these are commonly used to describe curves and shapes that are more naturally expressed in a radial and angular framework, such as spirals, circles, and sectors of circles.

Rectangular (Cartesian) Form

The rectangular, or Cartesian, form of an equation uses the traditional \(x\) and \(y\) coordinates that designate horizontal and vertical positions on a graph.

Converting from polar to rectangular coordinates involves using the connections between \(x\), \(y\), \(r\), and \(\theta\) to rewrite the polar equations into equivalent expressions with \(x\) and \(y\) only. The equivalent rectangular equation characterizes the same geometric shape on the Cartesian coordinate plane instead of the polar coordinate system.

Converting Polar Coordinates

To convert polar coordinates to the rectangular form, we apply trigonometric identities and the Pythagorean theorem. The fundamental relationships are \(x = r\cos\theta\) and \(y = r\sin\theta\), with \(r^2 = x^2 + y^2\) connecting the two forms.

When converting, we express 'r' and 'θ' in terms of 'x' and 'y'. For the original exercise, by substituting \(r\sin\theta\) with 'y', we reformulate the equation as \(r^2 = 2y\). Subsequently, replacing \(r^2\) with \(x^2 + y^2\), we obtain the final rectangular form \(x^2 + y^2 = 2y\), which is an equation in Cartesian coordinates representing the same curve defined by the original polar equation.