Problem 85 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\frac{4}{1-\cos \theta} $$

Step-by-Step Solution

Answer

The equation is already in polar coordinates: \ r = \frac{4}{1 - \cos \theta} \.

Step 1: Recall the conversion formulas

In polar coordinates, the relationships between rectangular coordinates \(x\) and \(y\) and polar coordinates \(r\) and \( \theta \) are \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Identify the given polar equation

The provided equation is already in polar form: \[ r = \frac{4}{1 - \cos \theta} \]

Step 3: Recognize the polar form structure

The equation \[ r = \frac{4}{1 - \cos \theta} \] represents a conic section with polar coordinates. For this specific form, it represents a parabola.

Step 4: Convert to rectangular coordinates (Optional)

To convert \[ r = \frac{4}{1 - \cos \theta} \] into rectangular coordinates, use the identities \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r = \sqrt{x^2 + y^2} \) as follows (though it might be complex in this case and unnecessary since the equation is already in polar form):1. Isolate \( \cos \theta \): \[ \cos \theta = \frac{x}{\sqrt{x^2 + y^2}} \]2. Substitute \( \cos \theta \) back into the original equation.

Key Concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to locate points in a plane using two perpendicular axes - typically labeled as the x-axis and y-axis. Each point on the plane is represented by a pair of numerical values \(x\) and \(y\). These values denote distances from the axes.
For example, a point \( (3, 4) \) indicates that the point is 3 units right from the y-axis and 4 units up from the x-axis.
Rectangular coordinates are very useful in a wide range of mathematical applications, including graphing equations and plotting data.

conversion formulas

Conversion formulas are mathematical equations used to translate points between different coordinate systems. In our case, we focus on converting between rectangular coordinates and polar coordinates.
Here are the key formulas to remember:

• From Polar to Rectangular: \(x = r \cos \theta\) and \(y = r \sin \theta\)
• From Rectangular to Polar: \(r = \sqrt{x^2 + y^2}\) and \( \theta = \arctan \left( \frac{y}{x} \right)\)

The parameter \(r\) represents the radius (or distance from the origin), and \( \theta \) represents the angle measured from the positive x-axis.

polar form

The polar form is another way of representing points in a plane, using a distance from a reference point (origin) and an angle from a reference direction (positive x-axis). This system is particularly useful for equations and problems involving circular or spherical shapes.

In polar coordinates \( (r, \theta) \), the parameter \(r\) indicates how far the point is from the origin, while \( \theta \) indicates the angle measured counterclockwise from the positive x-axis.

For example, converting the equation \( r = \frac{4}{1 - \cos \theta} \) to rectangular coordinates involves using the identities \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r = \sqrt{x^2 + y^2} \). However, in many cases, it is more practical to leave such equations in their polar form due to their simplicity in describing shapes like circles or parabolas.

Remember, understanding polar coordinates and their conversion to rectangular coordinates can make solving complex geometric problems much easier.