Problem 49 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=2 \cos \theta+2 \sin \theta$$

Step-by-Step Solution

Answer

The Cartesian equation is \((x-1)^2 + (y-1)^2 = 2\); it is a circle centered at \((1,1)\) with radius \(\sqrt{2}\).

Step 1: Understand the Polar Equation

We are given the polar equation \(r = 2\cos\theta + 2\sin\theta\). We need to express this equation in Cartesian coordinates \(x\) and \(y\). Remember that \(r\) represents the radius from the origin to a point, while \(\theta\) is the angle from the positive \(x\)-axis.

Step 2: Use Polar to Cartesian Conversion Formulas

Recall the conversion formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\). Additionally, \(r^2 = x^2 + y^2\). These will help us transform the equation into Cartesian coordinates.

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

Since \(r\cos\theta = x\) and \(r\sin\theta = y\), rewrite the equation \(r = 2\cos\theta + 2\sin\theta\) as \(r = 2\left(\frac{x}{r}\right) + 2\left(\frac{y}{r}\right)\), which simplifies to \(r = \frac{2x}{r} + \frac{2y}{r}\). Multiplying through by \(r\) to clear the fractions, we get \(r^2 = 2x + 2y\).

Step 4: Replace \(r^2\) with \(x^2 + y^2\)

Substitute \(r^2 = x^2 + y^2\) into the equation from Step 3: \[ x^2 + y^2 = 2x + 2y \] This is now a Cartesian equation.

Step 5: Identify the Graph

Rewrite the equation as: \[ x^2 - 2x + y^2 - 2y = 0 \] Complete the square for both \(x\) and \(y\): \[ (x-1)^2 - 1 + (y-1)^2 - 1 = 0 \]Adding the 1's to both sides, it becomes: \[ (x-1)^2 + (y-1)^2 = 2 \]This is the equation of a circle with center \((1,1)\) and radius \(\sqrt{2}\).

Key Concepts

Polar Coordinates

Polar coordinates are a method of expressing points in a two-dimensional plane through a radius and an angle. These coordinates are especially useful for problems involving rotation or circular motion. In polar coordinates, a point is represented as \((r, \theta)\):

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

The angle \(\theta\) is typically expressed in radians. Polar coordinates can offer a simpler approach sometimes, especially for equations involving circles or other curves that originate from a single point and radiate outward.
However, for other uses, like straight lines or more complex curves, cartesian coordinates might be simpler or preferable.

Cartesian Coordinates

Cartesian coordinates are a standard method for describing a position in a plane through two numerical coordinates that are distances from two intersected perpendicular axes, typically labeled as the x-axis and y-axis.

• The x-coordinate measures the distance to the point along the horizontal x-axis.
• The y-coordinate measures the distance to the point along the vertical y-axis.

Each point is represented by \((x, y)\). The Cartesian system is beneficial for solving algebraic equations and graphing linear relationships. In this exercise, we were tasked with converting a polar equation into a Cartesian form to facilitate understanding and to identify its geometrical representation on the Cartesian plane.

Circle Equation

The equation of a circle in the Cartesian coordinate system is derived from its definition: it is the set of all points that are equidistant from a fixed point, known as the center. The general form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius.

In the example provided, from converting the polar equation into Cartesian form, we identified the circle's equation as \((x-1)^2 + (y-1)^2 = 2\). Here, the center is located at \((1, 1)\) and the radius is \(\sqrt{2}\). This form makes it easier to visualize the exact size and location of the circle in a plane.