Problem 31 - Calculus Early Transcendentals

Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=2 \sin \theta+2 \cos \theta\)

Step-by-Step Solution

Answer

The equivalent Cartesian coordinate equation is \((x - 1)^2 + (y - 1)^2 = 2\), which represents a circle with center at \((1, 1)\) and a radius of \(\sqrt{2}\).

Step 1: Convert r to x and y coordinates

To convert the given equation from polar to Cartesian coordinates, we can substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the polar equation. Given equation: \(r = 2 \sin \theta + 2 \cos \theta\) Substituting the conversion formulas: \(r = 2\frac{y}{r} + 2 \frac{x}{r}\).

Step 2: Simplify the Equation

Now, we will simplify the equation to completely eliminate the polar variables: \(r^2 = 2y + 2x\) Since \(r^2 = x^2 + y^2\), we can write the equation as: \(x^2 + y^2 = 2x + 2y\). This is the converted Cartesian equation.

Step 3: Rearrange the Equation into a Recognizable Form

Rearrange the equation to a recognizable form to identify the curve: \(x^2 - 2x + y^2 - 2y = 0\) Now we can complete squares for x and y terms: \((x^2 - 2x + 1) + (y^2 - 2y + 1) = 2\) \((x - 1)^2 + (y - 1)^2 = 2\)

Step 4: Identify the Resulting Curve

The simplified equation is now in the form of a circle: \((x - h)^2 + (y - k)^2 = r^2\) where \((h,k)\) is the center and \(r\) is the radius. In this case, we have: \((x - 1)^2 + (y - 1)^2 = 2\) Therefore, the center of the circle is \((1, 1)\) and the radius is \(\sqrt{2}\). This is the description of the resulting curve in Cartesian coordinates.

Key Concepts

Polar Coordinates

Polar coordinates provide a method of representing points in a plane using a radius and an angle relative to a reference direction. Specifically, a point in polar coordinates is denoted as \(r, \theta\), where \(r\) represents the distance from the point to the origin (often called the radial coordinate) and \(\theta\) indicates the counterclockwise angle from the positive x-axis to the point (the angular coordinate).

Understanding polar coordinates is crucial when dealing with problems that have a natural circular symmetry, where using polar coordinates can simplify the equations and calculations involved.

Cartesian Coordinates

Cartesian coordinates, also known as rectangular coordinates, are the most well-known coordinate system, using two or three perpendicular axes to determine the position of points in a plane or space. In two dimensions, each point is specified by an ordered pair \(x, y\), where \(x\) is the horizontal distance from the y-axis, and \(y\) is the vertical distance from the x-axis.

Cartesian coordinates are extensively used in algebra and calculus because they allow for straightforward representation of geometric shapes, easy calculation of distances, and simple algebraic manipulations.

Equation Conversion

Converting equations between polar and Cartesian coordinates is a fundamental technique in mathematics, especially when analyzing curves and shapes. The conversion involves using the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\), which are derived from the definitions of sine and cosine in right-angled triangles. By substituting these into a polar equation, one can rewrite the equation in terms of \(x\) and \(y\) alone, effectively converting the equation to Cartesian coordinates.

This method also works the other way around, allowing for conversion from Cartesian to polar coordinates by using \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\) or appropriate variations taking into account the quadrant of the point.

Circle Equation

In Cartesian coordinates, a circle's equation typically has the form \(x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) is the center of the circle and \(r\) is its radius. This form is derived from the Pythagorean theorem applied to a right-angled triangle formed by a radius of the circle, which has its vertex at the origin of the coordinate system and its hypotenuse passing through a point on the circle.

To identify the circle described by a given equation, one must often complete the square for the \(x\) and \(y\) terms, which can result in recognizing the center and the radius squared. This is critical when converting polar equations to Cartesian form since the process can obfuscate the underlying geometric properties of the curve.