Problem 49 - Contemporary Precalculus

Question

Find a rectangular equation that is equivalent to the given polar equation. $$r=2 \sin \theta$$

Step-by-Step Solution

Answer

The rectangular equation representing the curve of the polar equation $$r = 2\sin\theta$$ is $$y^2 - 2y + x^2 = 0$$.

Step 1: Identify the given polar equation and conversion formulas

The given polar equation is $$r=2\sin\theta$$. We will use the following conversion formulas to convert this equation into a rectangular equation: $$x = r\cos\theta$$ $$y = r\sin\theta$$ $$r = \sqrt{x^2 + y^2}$$

Step 2: Convert sin\(\theta\) and cos\(\theta\) in terms of x and y

Since $$x = r\cos\theta$$, we can solve for $$\cos\theta$$: $$\cos\theta = \frac{x}{r}$$ And since $$y = r\sin\theta$$, we can solve for $$\sin\theta$$: $$\sin\theta = \frac{y}{r}$$

Step 3: Replace sin\(\theta\) with its rectangular form in the given polar equation

Now, replace sin\(\theta\) in the given polar equation: $$r = 2\sin\theta \to r=2\frac{y}{r}$$

Step 4: Replace r with its rectangular form

Now, replace $$r$$ with its rectangular form in the equation from Step 3: $$\sqrt{x^2 + y^2} = 2\frac{y}{\sqrt{x^2 + y^2}}$$

Step 5: Simplify the equation and solve for y

To simplify the equation, multiply both sides by the common denominator $$\sqrt{x^2 + y^2}$$: $$(x^2 + y^2) = 2y$$ Now, solve for $$y$$ to get the rectangular equation: $$y^2 - 2y + x^2 = 0$$ This is the rectangular equation that represents the same curve as the original polar equation, $$r=2\sin\theta$$.

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to locate points in a plane. Rather than using horizontal and vertical axes, they use a distance and an angle.

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

These coordinates are particularly useful in scenarios involving circular motion or when dealing with periodic functions. Understanding polar coordinates is crucial because they simplify many mathematical problems involving curves and circles.

Conversion Formulas

Conversion between polar and rectangular coordinates is common in mathematics. The ability to switch between these systems allows us to analyze problems from different perspectives. Here's how the conversion works:

• To convert from polar to rectangular, use the formulas: \[ x = r \cos \theta \] \[ y = r \sin \theta \] These equations give us the rectangular coordinates using \(r\) and \(\theta\).
• To switch back, the formulas are: \[ r = \sqrt{x^2 + y^2} \] \[ \theta = \tan^{-1}(\frac{y}{x}) \]

Employing these conversions helps in translating problems to a more familiar rectangular form when needed.

Trigonometric Identities

Trigonometric identities are essential tools that help us rewrite expressions involving trigonometric functions. They reveal relationships between angles and sides of a triangle.

• The primary identities you'll often use include: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This is known as the Pythagorean identity.
• Other identities involve ratios and reciprocal relationships: \[ \tan \theta = \frac{\sin \theta}{\cos \theta}\]

Using these identities enables simplification of equations during conversion, making problem-solving easier and more straightforward.

Rectangular Coordinates

Rectangular coordinates are the standard way to describe points in a plane, using horizontal and vertical distances.

• Each point \((x, y)\) denotes its distance from the y-axis (x) and the x-axis (y).
• This system is prevalent in graphing and is used in a wide range of applications.

By converting polar equations like \(r = 2 \sin \theta\) into rectangular form, we can more easily interpret and analyze the relationship of the equations on a standard x-y coordinate plane, which is familiar to most students.