Problem 50 - 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. $$r=2 \cos \theta-\sin \theta$$

Step-by-Step Solution

Answer

The Cartesian equation is a circle centered at \((1, -\frac{1}{2})\) with radius \(\frac{\sqrt{5}}{2}\).

Step 1: Understanding Polar to Cartesian Conversion

In polar coordinates, any point is represented by \((r, \theta)\) where \(r\) is the radius, and \(\theta\) is the angle. To convert this to Cartesian coordinates \((x, y)\), we use the relationships: \(x = r \cos \theta\) and \(y = r \sin \theta\). Additionally, \(r^2 = x^2 + y^2\) and \(\tan \theta = \frac{y}{x}\).

Step 2: Multiply Both Sides by r

The given equation is \(r = 2 \cos \theta - \sin \theta\). To start the conversion, we multiply both sides by \(r\), yielding: \(r^2 = r (2 \cos \theta - \sin \theta)\).

Step 3: Substitute Polar to Cartesian

Substitute \(r^2 = x^2 + y^2\), \(r \cos \theta = x\), and \(r \sin \theta = y\) into the equation: \[x^2 + y^2 = 2x - y\]

Step 4: Rearrange into Standard Form

Rearrange and simplify the equation to express it in a standard form:\[x^2 + y^2 - 2x + y = 0\]

Step 5: Complete the Square

To identify the graph, complete the square for both x and y terms:For x-term: \((x^2 - 2x) \rightarrow (x - 1)^2 - 1\)For y-term:\((y + y) \rightarrow (y + \frac{1}{2})^2 - \frac{1}{4}\)The equation becomes:\[(x - 1)^2 + (y + \frac{1}{2})^2 = \frac{5}{4}\]

Step 6: Identify the Graph

The equation \((x - 1)^2 + (y + \frac{1}{2})^2 = \frac{5}{4}\) represents a circle with center at \((1, -\frac{1}{2})\) and radius \(\frac{\sqrt{5}}{2}\).

Key Concepts

Polar Coordinates

Polar coordinates are a method of representing points in a plane using two values: the radius and the angle. The radius, denoted as \(r\), is the distance from the origin to the point. The angle, denoted as \(\theta\), is measured from the positive x-axis in a counter-clockwise direction to the line connecting the origin and the point. This system is particularly useful in scenarios involving circular paths or rotational movements.

• The radius \(r\) defines how far the point is from the center.
• The angle \(\theta\) indicates the direction relative to the origin.

The conversion from polar to Cartesian coordinates involves translating this radial and angular measurement into a position defined by x and y coordinates on a grid. You can think of polar coordinates as a natural fit for problems involving curves around a central point, such as orbits or spirals.

Cartesian Coordinates

Cartesian coordinates provide a straightforward way of identifying locations on a plane through two perpendicular axes: the x-axis and the y-axis. Any point on a plane can be described by an ordered pair \((x, y)\), where \(x\) represents the horizontal distance from the origin and \(y\) represents the vertical distance. This system is integral to algebra and calculus, often serving as the backbone for defining functions and graphs.

• The x-coordinate determines position along the horizontal axis.
• The y-coordinate determines position along the vertical axis.

This method is intuitive for plotting and working with linear paths and rectangular shapes, and it underpins most analytical geometry problems. Understanding Cartesian coordinates is crucial for any conversion from polar coordinates, as each point has a unique pair of Cartesian coordinates mapping to it.

Equation Conversion

Converting polar equations to Cartesian equations involves substituting polar coordinates with their Cartesian counterparts. This process enables us to represent and solve problems in the more familiar Cartesian framework, which simplifies the analysis of geometric shapes and functional relationships.Here's a quick look at how this works:- Given polar equations like \(r = a\cos\theta + b\sin\theta\), the conversion relies on using: - \(x = r \cos \theta\) - \(y = r \sin \theta\) - \(r^2 = x^2 + y^2\)Making these substitutions allows for the transformation of a polar equation into a Cartesian one, revealing the underlying geometric shape, such as circles, ellipses, or lines. For instance, in the example \(r = 2 \cos \theta - \sin \theta\), these substitutions lead to the Cartesian form \((x - 1)^2 + (y + \frac{1}{2})^2 = \frac{5}{4}\), describing a circle. This conversion not only aids in visualization but also assists in applying calculus and algebraic methods to analyze and solve problems.