Problem 96 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. Prove that the graph of each polar equation is a circle, and find its center and radius. (a) \(r=a \sin \theta, a \neq 0\) (b) \(r=b \cos \theta, b \neq 0\) (c) \(r=a \sin \theta+b \cos \theta, a \neq 0\) and \(b \neq 0\)

Step-by-Step Solution

Answer

All given polar equations sketch as circles with calculated centers and radii.

Step 1: Analyze Equation Part (a)

Consider the polar equation \( r = a \sin \theta \). In polar coordinates, this form corresponds to a circle. To prove, transform it into Cartesian coordinates using the relations \( x = r \cos \theta \) and \( y = r \sin \theta \). Thus, \( y = a \sin \theta \sin \theta \) simplifies to \( y = a \sin^2 \theta \), or \( y = \frac{a}{2}(1 - \cos(2\theta)) \) after using trigonometric identities, which centers the circle on the y-axis. Further calculations yield \( x^2 + (y - \frac{a}{2})^2 = (\frac{a}{2})^2 \). Hence the center is \((0, \frac{a}{2})\) with radius \(\frac{a}{2}\).

Step 2: Analyze Equation Part (b)

Consider the polar equation \( r = b \cos \theta \). Transform it to Cartesian coordinates using \( x = r \cos \theta \) and \( y = r \sin \theta \). This yields \( x = b \cos \theta \cos \theta \), simplifying to \( x = \frac{b}{2}(1 + \cos(2\theta)) \) after using trigonometric identities, effectively centering the circle along the x-axis. After further manipulation, we find \( (x - \frac{b}{2})^2 + y^2 = (\frac{b}{2})^2 \). The center is thus \((\frac{b}{2}, 0)\) with radius \(\frac{b}{2}\).

Step 3: Analyze Equation Part (c)

For the equation \( r = a\sin\theta + b\cos\theta \), first express it in Cartesian form. Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \), resulting in equations \( a\sin\theta = y \) and \( b\cos\theta = x \). Adding these and complete the square gives the circle's equation as \((x - \frac{b}{2})^2 + (y - \frac{a}{2})^2 = \left(\frac{a^2 + b^2}{2}\right)\). Thus, its center is \((\frac{b}{2},\frac{a}{2})\) with radius \(\frac{\sqrt{a^2 + b^2}}{2}\).

Key Concepts

Polar Coordinates

Polar coordinates offer an exciting way to represent points in a plane by specifying the distance from a reference point and the angle from a reference direction. Polar coordinates are denoted as \((r, \theta)\), where:

• \(r\) is the radius or the distance from the origin.
• \(\theta\) is the angle measured from the positive x-axis.

Using polar coordinates can simplify problems involving symmetry or rotation. In contrast to Cartesian coordinates which use a grid system of x and y axes, polar coordinates emphasize the direction and magnitude of points.
To convert between polar and Cartesian coordinates:

• Use \(x = r \cos \theta\) to find the x-coordinate.
• Use \(y = r \sin \theta\) to find the y-coordinate.

By mastering this conversion, you can easily shift between the two systems when analyzing equations or graphs.

Cartesian Coordinates

Cartesian coordinates provide a straightforward method to represent points on a plane using a pair of numbers. The system uses horizontal and vertical axes, known as the x and y axes. Each point is expressed as \((x, y)\):

• \(x\) is the horizontal distance from the y-axis.
• \(y\) is the vertical distance from the x-axis.

This rectangular coordinate system is particularly useful for describing linear equations, parabolas, and other algebraic expressions.
When working with polar equations, converting to Cartesian coordinates can reveal detailed geometric properties. For example, transforming polar equations into Cartesian form can show that some polar graphs are actually circles, even when they first appear complex. This process involves using the relations \(x = r \cos \theta\) and \(y = r \sin \theta\), making it easier to manipulate and analyze the equations.

Trigonometric Identities

Trigonometric identities are essential tools for simplifying complex mathematical expressions involving angles. These identities are equations that hold for any value of the involved variable. Some commonly used ones include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Double Angle Formula: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta \)
• Angle Sum Formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

Trigonometric identities are particularly useful when transforming polar equations into Cartesian forms. They help break down expressions like \( r = a \sin \theta \) or \( r = b \cos \theta \) into simpler forms, revealing geometric shapes like circles. Mastery of these identities allows you to manipulate and reduce equations, making cross-referencing between polar and Cartesian forms more intuitive.

Circle Center and Radius

Understanding a circle's center and radius is key when analyzing polar equations. A circle is uniquely determined by:

• Its center, a point expressed as \((h, k)\).
• Its radius, a distance noted as \(r\).

In polar coordinates, some equations naturally describe circles. For instance, with \( r = a \sin \theta\) and \(r = b \cos \theta\), transformation into Cartesian coordinates might reveal simple forms like:

• Equation: \((x - h)^2 + (y - k)^2 = r^2\)
declaring a circle centered at \((h, k)\) with radius \(r\).

By identifying the circle's center and its radius from polar equations, you gain a clear visualization of its graph. This approach simplifies problems, making it less about complex algebra and more about geometric understanding.