Question

For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), so the polar equation \(r=f(\theta)\) is now written in parametric form.Find the points on the interval \(-\pi \leq \theta \leq \pi\) at which the cardioid \(r=1-\cos \theta\) has a vertical or horizontal tangent line.

Short answer

The cardioid has horizontal tangents at \( \theta = 0 \) and \( \pi \); vertical tangents at \( \theta = \pi/2 \) and \( -\pi/2 \).

Step-by-step solution

Express x(θ) and y(θ)

Let the polar equation be \( r = 1 - \cos \theta \). Write the parametric equations: \( x(\theta) = (1 - \cos \theta) \cos \theta \) and \( y(\theta) = (1 - \cos \theta) \sin \theta \).

Find dx/dθ and dy/dθ

Differentiate the parametric equations with respect to \( \theta \):- \( \frac{dx}{d\theta} = -\sin \theta \cdot (1 - \cos \theta) - \cos \theta \cdot \sin \theta \) \[ = \sin^2 \theta - \sin \theta \] - \( \frac{dy}{d\theta} = \cos \theta \cdot (1 - \cos \theta) - \sin \theta \cdot \sin \theta \) \[ = \cos \theta - \cos^2 \theta - \sin^2 \theta \] Use trigonometric identity \( -\cos^2 \theta - \sin^2 \theta = -1 \).

Solve for Horizontal Tangents \((\frac{dy}{d\theta} = 0)\)

For a horizontal tangent, set \( \frac{dy}{d\theta} = 0 \):\[ \cos \theta(1 - \cos \theta) - 1 = 0 \]Solving yields \( \cos \theta = 1 \). Check solutions, noting \( \theta = 0 \) and repeat to verify via the unit circle.

Solve for Vertical Tangents \((\frac{dx}{d\theta} = 0)\)

For a vertical tangent, set \( \frac{dx}{d\theta} = 0 \):\[ \sin^2 \theta - \sin \theta = 0 \]Factoring to \( \sin \theta(\sin \theta - 1) = 0 \), find \( \sin \theta = 0 \) or \( \sin \theta = 1 \). These occur at \( \theta = 0, \pi, \pi/2, -\pi/2 \).

Identify Points of Tangency

Returning to the initial cardioid function:- Verify critical \( \theta \) solutions are within \( [-\pi, \pi] \).- For \( \theta = 0 \), \( \pi \): horizontal since \( \cos \theta = 1 \).- For \( \theta = \pi/2, -\pi/2 \): vertical since \( \sin \theta = 1 \text{ or } -1 \).

Key concepts

Parametric Equations

Parametric equations are a way to represent curves using parameters. Instead of defining a relationship directly between two variables like x and y, you express each variable as a function of another variable, usually denoted as \( \theta \) or \( t \).

In the context of polar coordinates, converting a polar equation \( r = f(\theta) \) into parametric form involves expressing x and y in terms of \( \theta \) rather than each other.

Here's how it's done:

• For \( x(\theta) = r \cos \theta = f(\theta) \cos \theta \)
• For \( y(\theta) = r \sin \theta = f(\theta) \sin \theta \)

By using these transformations, you essentially create functions for x and y with respect to \( \theta \), allowing you to work with complex curves more easily. This representation is particularly useful when dealing with curves that have varying radii, like cardioids or spirals.

Understanding parametric equations helps you analyze and graph curves that are otherwise difficult to describe using conventional Cartesian coordinates.

Tangent Lines

In mathematics, tangent lines are straight lines that touch a curve at a single point, showing the direction and rate of change of the curve at that exact location. Tangent lines are crucial for understanding the behavior of curves and for defining concepts like slopes and derivatives.

For parametric equations in polar coordinates, the concept of tangent lines translates into finding the derivatives \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \). The slope of the tangent line is given by the ratio:

• \( \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \)

This ratio tells you the slope at any given point \( (x(\theta), y(\theta)) \) on the curve.

Horizontal tangent lines occur when \( \frac{dy}{d\theta} = 0 \) because the y-component is not changing, while vertical tangent lines occur when \( \frac{dx}{d\theta} = 0 \) because the x-component is not changing. By setting these derivatives to zero, one can find the specific points where these tangent lines occur on the curve.

Cardioid

A cardioid is a special type of curve that resembles the heart shape and can be represented through polar coordinates. The general equation for a cardioid in polar form is \( r = a(1 - \cos \theta) \) or \( r = a(1 + \cos \theta) \). For the given exercise, the specific cardioid is expressed by the equation \( r = 1 - \cos \theta \).

Cardioids are prominent in polar coordinates due to their distinct shape and properties, such as symmetry and loops. They're often utilized in signal processing and acoustics.

In terms of tangency, analyzing cardioids involves finding points on the curve where the tangent lines are either vertical or horizontal, highlighting the curve's geometric features.

• Horizontal tangents occur where \( \cos \theta = 1 \), leading to a constant radius.
• Vertical tangents occur where \( \sin \theta = 1 \) or \( \sin \theta = -1 \), indicating a sharp change in direction.

Identifying these points in the cardioid helps in understanding the directional changes, which are crucial for graphing and interpreting the curve’s behavior.