Problem 12 - Algebra and Trigonometry

Question

True or False A cardioid passes through the pole.

Step-by-Step Solution

Answer

True

Step 1: Understand the Definition of a Cardioid

A cardioid is a heart-shaped curve which can be described by the polar equation \( r = a(1 + \cos(\theta)) \) or \( r = a(1 - \cos(\theta)) \). The term 'pole' in polar coordinates refers to the origin \( (0,0) \).

Step 2: Substitute \( \theta \) with an Appropriate Value

Substitute \( \theta = \pi \) in the equation \( r = a(1 + \cos(\theta)) \). This substitution is chosen because \cos(\pi) = -1\.

Step 3: Evaluate the Equation

Evaluating \( r \) at \( \theta = \pi\): \[ r = a(1 + \cos(\pi)) = a(1 - 1) = a \cdot 0 = 0 \. \] This shows that \( r = 0 \) when \( \theta = \pi \), which means the cardioid passes through the pole.

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing the position of a point in a plane. Unlike Cartesian coordinates which use \((x, y)\) pairs, polar coordinates use \((r, \theta)\) pairs. In this system, \(r\) represents the distance from a central point called the pole (similar to the origin in Cartesian coordinates), and \(\theta\) represents the angle from a reference direction.

To better understand this, imagine a point on a circular grid. The radius \(r\) tells you how far the point is from the center, while the angle \(\theta\) tells you in which direction you need to go to get to the point. This is particularly useful in problems involving circular or spiral patterns.

For instance, let's consider a point at \(r = 5\) and \(\theta = \frac{\text{π}}{3}\). This means the point is 5 units away from the origin and forms an angle of \(\frac{\text{π}}{3}\) radians (or 60 degrees) from the positive x-axis. The pole is essentially the anchor point, and the angle helps you navigate around the circular grid.

Cardioid

A cardioid is a special type of curve that looks like a heart shape. It can be defined using polar coordinates and has the general equations \(r = a(1 + \text{cos}(\theta))\) or \(r = a(1 - \text{cos}(\theta))\). Here, 'a' is a constant that determines the size of the cardioid.

The distinctive heart shape of a cardioid is due to its unique mathematical properties. When graphed, a cardioid has a cusp at the origin (the pole) and is symmetric about the x-axis if defined with cosine. This symmetry gives it its heart-like appearance, with one lobe pointing outward and the other inward toward the pole.

One crucial point to note is that cardioids always pass through the origin or pole. This can be shown by substituting particular values into the equation. For instance, if we substitute \(\theta = \text{π}\) into \(r = a(1 + \text{cos}(\theta))\), we get \(r = a(1 - 1) = 0\), proving that the curve intersects the origin.

Polar Equations

Polar equations are equations that express relationships between the radius \(r\) and the angle \(\theta\) in polar coordinates. These equations are especially useful for describing curves that are naturally circular or radial, such as spirals, roses, and cardioids.

A polar equation typically has the form \(r = f(\theta)\), where \(f(\theta)\) is a function that defines the distance \(r\) as a function of the angle \(\theta\). For instance, \(r = a(1 + \text{cos}(\theta))\) is a polar equation for a cardioid.

Polar equations can sometimes be converted to Cartesian coordinates for easier visualization or manipulation. For example, the cardioid equation \(r = a(1 + \text{cos}(\theta))\) can be transformed into Cartesian coordinates using the relationships \(x = r \text{cos}(\theta)\) and \(y = r \text{sin}(\theta)\), though the resulting equation might be more complex.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in the study of polar coordinates and polar equations. These functions help describe the relationships between the angles and distances in circular motion or oscillations.

Cosine and sine functions are especially important in defining cardioid equations. For a cardioid described by \(r = a(1 + \text{cos}(\theta))\), the cosine function determines how the radius changes with the angle \(\theta\). When \(\theta = 0\), \(\text{cos}(0) = 1\), making \(r = 2a\). When \(\theta = \text{π}\), \(\text{cos}(\text{π}) = -1\), resulting in \(r = 0\) and showing the curve passes through the origin.

The variation of \(\text{cos}(\theta)\) between -1 and 1 generates the distinct 'heart' shape of the cardioid. Trigonometric functions also help identify other key properties of curves, like symmetry and periodicity, making them invaluable tools in the study of polar coordinates and equations.