Problem 28 - Calculus Early Transcendentals

Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=\cot \theta \csc \theta\)

Step-by-Step Solution

Answer

Question: Convert the given polar equation to Cartesian coordinates and describe the shape of the curve: \(r=cot \theta \csc \theta\). Answer: The Cartesian form of the given polar equation is \(y^2 = x^2\). The shape of the curve is a pair of straight lines, \(y = \pm x\), which pass through the origin and form a \(45^\circ\) angle with the positive and negative x-axis, respectively. This curve is known as a pair of "rectangular hyperbolas".

Step 1: Rewrite the given equation using trigonometric identities

We have the polar equation \(r=\cot \theta \csc \theta\). Recall that \(\cot \theta =\frac{\cos \theta}{\sin\theta}\) and \(\csc \theta=\frac{1}{\sin\theta}\). Let's rewrite the equation using these identities: \(r=\frac{\cos\theta}{\sin\theta} \cdot \frac{1}{\sin\theta}\) Now, simplifying the equation, we get: \(r=\frac{\cos\theta}{\sin^2 \theta}\)

Step 2: Substitute \(x\) and \(y\) with polar relationships

Since the equations \(x=r\cos \theta\) and \(y=r\sin \theta\) describe the transition from polar to Cartesian coordinates, we will use these equations to substitute the polar coordinates with Cartesian coordinates: First, let's observe that \(\cos \theta=\frac{x}{r}\) and \(\sin \theta=\frac{y}{r}\). Now, replacing the expression for \(r\) in the equation using these substitutions: \(r=\frac{x}{\left(\frac{y}{r}\right)^2}\)

Step 3: Solve for the remaining polar variable

Now, we need to eliminate the remaining polar variable, \(r\). Let's start by simplifying the equation: \(r=\frac{rx^2}{y^2}\) Dividing both sides by \(x^2\): \(\frac{r}{x^2}=\frac{r}{y^2}\) To get rid of the \(r\), we can multiply both sides by \(\frac{x^2}{r}\): \(1=\frac{x^2}{y^2}\) The obtained equation is in Cartesian coordinates.

Step 4: Describe the resulting curve

The Cartesian equation is: \(1=\frac{x^2}{y^2}\) or, \(y^2=x^2\) Taking square root on both sides, we have: \(y=\pm x\) This represents two straight lines that pass through the origin with slopes of \(1\) and \(-1\). These lines intersect at a right angle, forming a \(45^\circ\) angle with the positive x-axis and negative x-axis, respectively. This curve is known as a pair of "rectangular hyperbolas".

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points on a plane using a distance from a reference point and an angle from a reference direction. This system is especially useful in scenarios where the relationship between two points is more intuitive when defined by a direction and a distance, rather than rectangular coordinates.

• The reference point is called the pole (similar to an origin in Cartesian coordinates).
• The direction is typically given as an angle \(\theta\) measured from a reference direction, often the positive x-axis.
• The distance from the pole is denoted by \(r\), indicating how far the point is from the origin.

In the exercise, the polar equation was given as \(r = \cot \theta \csc \theta\). To convert to Cartesian coordinates, it's crucial to use specific relationships, such as \(x = r \cos \theta\) and \(y = r \sin \theta\). These allow us to translate problems formulated in polar terms into the familiar Cartesian framework.

Trigonometric Identities

Trigonometric identities are fundamental equations that express relationships between trigonometric functions. They are essential tools in simplifying and transforming complex trigonometric expressions, such as converting polar equations to Cartesian coordinates.

• Basic Identities: such as \(\sin^2 \theta + \cos^2 \theta = 1\), are foundational.
• Reciprocal Identities: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), often help in simplifying expressions.
• Quotient Identities: like \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), are also crucial.

In this exercise, the identities \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\) were used to re-express the polar equation as \(r = \frac{\cos \theta}{\sin^2 \theta}\). This transformation is the first step towards converting the equation into Cartesian form.

Rectangular Hyperbolas

Rectangular hyperbolas are a type of confocal conic sections defined by certain equations in Cartesian coordinates. A key feature of these curves is that they consist of two intersecting lines at right angles, hence the term "rectangular."

• The general form of the equation is \(xy = c^2\), where \(c\) is a constant.
• In the solution, the equation \(y^2 = x^2\) represents two straight lines \(y = x\) and \(y = -x\), which intersect at the origin.
• This pair of lines is considered a degenerate case of a rectangular hyperbola, as they do not form a closed curve but instead, form the axes themselves.

These lines demonstrate the geometric nature of such hyperbolas, highlighting the importance of understanding not just the algebraic manipulation but also the geometric interpretation of the resulting equation. The exercise showcases how different coordinate systems can offer varying insights into familiar geometric shapes.