Problem 32 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Convert the polar equation \(\sin \theta = |\cos \theta|\) to Cartesian coordinates and describe the resulting curve. Answer: The Cartesian equation is \(y = |x|\), which represents a V-shaped curve, called the "absolute value function," symmetric with respect to the y-axis.

Step 1: Write down the given equation

The equation given is \(\sin \theta = |\cos \theta|\) in polar coordinates.

Step 2: Convert \(\sin \theta \) and \(\cos \theta \) to Cartesian coordinates

In order to convert the given equation to Cartesian coordinates, we recall that in polar coordinates, \(x = r\cos \theta\) and \(y = r\sin \theta\). We can then find \(\sin \theta\) and \(\cos \theta\) as follows: \(\sin \theta = \frac{y}{r}\) \(\cos \theta = \frac{x}{r}\)

Step 3: Replace the \(\sin \theta\) and \(\cos \theta\) by their Cartesian form

Substituting \(\sin \theta\) and \(\cos \theta\) with their Cartesian forms in the given equation, we get: \(\frac{y}{r} = |\frac{x}{r}|\)

Step 4: Simplify the equation

Since both sides of the equation have \(r\) in the denominator, we can multiply both sides by \(r\) to remove the fractions: \(y = |x|\)

Step 5: Describe the resulting curve

The Cartesian equation \(y = |x|\) represents two lines: 1. When \(x \geq 0\), the equation is \(y = x\). This is a straight line passing through the origin with a slope of \(1\) and in the first and second quadrants. 2. When \(x < 0\), the equation is \(y = -x\). This is a straight line passing through the origin with a slope of \(-1\) and in the third and fourth quadrants. Together, these two lines form a V-shaped curve, called the "absolute value function," symmetric with respect to the y-axis.

Key Concepts

Polar Coordinates

In mathematics, polar coordinates provide a way to describe a point in the plane using a combination of distance and angle rather than traditional x and y coordinates. Imagine the graph centered at the origin, where every point is described by its radial distance \( r \) from the origin and the angle \( \theta \) that it forms with a reference direction, typically the positive x-axis.

Here's a more detailed breakdown of how polar coordinates work:

• \( r \): This represents the distance between the point and the origin. It is always a non-negative value, \( r \geq 0 \).
• \( \theta \): This specifies the angle between the line connecting the point to the origin and the positive x-axis. Angles in polar coordinates can be measured in degrees or radians.

The conversion from polar to Cartesian coordinates involves using the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). The given equation \( \sin \theta = |\cos \theta| \) in polar form shows the interrelationship between the vertical position \( y \) and the horizontal span \( x \) when described with these coordinates. Converting polar equations to Cartesian forms often simplifies understanding and visualization of curves.

Absolute Value Function

The absolute value function is a fundamental component in algebra that represents the magnitude of a number, regardless of its sign. In simple terms, the absolute value of a number \( x \) is denoted as \( |x| \) and is defined as:

• \( x \) if \( x \geq 0 \)
• \(-x \) if \( x < 0 \)

This function transforms negative inputs into positive outputs, effectively creating a distance from zero for any real number. The equation \( y = |x| \) can be visualized as a V-shaped graph:

• For non-negative \( x \), the line is \( y = x \).
• For negative \( x \), the line is \( y = -x \).

Thus, it consists of two linear pieces that intersect at the origin. The symmetry of this function is along the y-axis, making it not only a mathematical tool but also a model for real-world phenomena, like measuring deviations from a reference point without regard to direction.

Curve Description

Describing curves in mathematics often involves transforming and graphically exploring different sets of equations. The transformation and description of curves play a crucial role in visualizing mathematical concepts. In the given problem, translating the polar equation \( \sin \theta = |\cos \theta| \) into Cartesian coordinates yields the equation \( y = |x| \).

Let's break down this Cartesian curve:

• The curve is V-shaped and consists of two straight lines: \( y = x \) when \( x \geq 0 \) and \( y = -x \) when \( x < 0 \).
• These two lines intersect at the origin, (0,0), forming an angle of 90 degrees between them.
• This shape is symmetric about the y-axis due to its reliance on the absolute value, which mirrors any changes about the y-axis.

Describing curves accurately allows for better understanding of graphs and their properties, a pivotal aspect in mathematics. Curve analysis extends to various fields, helping describe physical phenomena, relationships in economics, and more. By mastering the conversion between coordinate systems and understanding the resulting curve properties, students enhance their abilities to interpret and apply mathematical concepts effectively.