Problem 57 - Precalculus Mathematics for Calculus

Question

Show that the graph of \(r=a \cos \theta+b \sin \theta\) is a circle, and find its center and radius.

Step-by-Step Solution

Answer

The graph is a circle with center \((a/2, b/2)\) and radius \(\sqrt{a^2 + b^2}/2\).

Step 1: Understand Polar Equation

The given polar equation is \(r = a \cos \theta + b \sin \theta\). Our goal is to demonstrate that this equation represents a circle and to find its center and radius.

Step 2: Convert to Cartesian Coordinates

To convert the polar equation to Cartesian coordinates, use the relations \(x = r \cos \theta\) and \(y = r \sin \theta\). We also know that \(r = \sqrt{x^2 + y^2}\).

Step 3: Express \(x\) and \(y\) in Terms of \(\theta\)

Substitute the polar expressions into the equation: \(r = a \cos \theta + b \sin \theta\). With \(x = r \cos \theta\) and \(y = r \sin \theta\), \(x^2 + y^2 = (a \cos \theta + b \sin \theta)^2\).

Step 4: Simplify the Equation

Expand and simplify the equation: \(x^2 + y^2 = (a^2 \cos^2 \theta + 2ab \cos \theta \sin \theta + b^2 \sin^2 \theta)\). Observing trigonometric identities, \(\cos^2 \theta + \sin^2 \theta = 1\), we simplify to find a representation in Cartesian form.

Step 5: Convert Equation to a Circle

Apply the identities and rearrange the equation to complete the square. This should give us a standard Cartesian form of the circle, \((x - a/2)^2 + (y - b/2)^2 = (\sqrt{a^2 + b^2}/2)^2\), showing the center is \((a/2, b/2)\) and radius \(\sqrt{a^2 + b^2}/2\).

Step 6: Verify the Circle Equation

Ensure the equation derived is consistent with the known Cartesian form for a circle, \((x - h)^2 + (y - k)^2 = r^2\), noting that \(h = a/2\), \(k = b/2\), and the radius \(r = \sqrt{a^2 + b^2}/2\) confirms it is a circle.

Key Concepts

cartesian coordinates

Cartesian coordinates are a way to describe any point on a plane using a pair of numerical values. These values reflect the point's distance from two fixed, perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). In essence, every point is identified by a pair of numbers, \(x, y\), where \x\ is the horizontal distance and \y\ is the vertical distance from the origin (0,0).

The beauty of Cartesian coordinates lies in their ability to simplify complex geometric problems by translating them into algebraic equations. For example, conversions from polar (circular) to Cartesian coordinates involve expressions like \(x = r \cos \theta\) and \(y = r \sin \theta\). These equations help us understand and visualize curves and shapes in a rectangular plane, making Cartesian coordinates a versatile tool for mathematicians and engineers alike.

circle equation

A circle is a set of points equidistant from a center point. In Cartesian coordinates, the equation of a circle with center \(h, k\) and radius \r\ is given by the standard form: \((x-h)^2 + (y-k)^2 = r^2\).

This equation arises from the belief that every point on the circle lies a fixed radius away from the center. It allows us to plot the circle easily on a Cartesian plane by identifying the circle’s center and the extent of its reach.

To identify a circle in another form of an equation (like the one in the polar plane), one might have to rearrange the terms and complete the square to achieve this standard Cartesian form. Understanding the circle equation is crucial in solving geometry problems, especially those involving loci of points that satisfy certain conditions.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the given variables. They are essential in simplifying expressions and solving equations in trigonometry.

In the context of converting polar to Cartesian coordinates, some key identities are:

• \( \cos^2 \theta + \sin^2 \theta = 1\): This identity shows that the sum of the squares of sine and cosine of an angle equals 1. It is fundamental in finding relations in trigonometric equations.
• \( 1 - \cos^2 \theta = \sin^2 \theta\): This results by rearranging the basic identity, useful in simplifying terms.
• Double-angle identities and product-to-sum identities can also aid in converting and analyzing equations.

Recognizing these identities helps simplify expressions during conversions between systems of coordinates and aids in deriving specific geometric properties in equations.

convert polar to cartesian

Converting polar coordinates to Cartesian coordinates involves expressing a point defined by its distance and angle (r, \(\theta\)) in terms of its x and y coordinates, which are more intuitive for graphing on a plane.

The connection between the two systems uses the relations:

• \( x = r \cos \theta \) to find the horizontal component.
• \( y = r \sin \theta \) to determine the vertical component.

These transformations enable easier manipulation and understanding of figures originally presented in polar form. Once converted, we can use algebraic techniques to analyze geometric shapes like circles. In our problem, starting with a polar equation \( r = a \cos \theta + b \sin \theta \), these transformations help reveal the circle's characteristics, such as its center and radius in the Cartesian plane.