Problem 18 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$r=-1$$

Step-by-Step Solution

Answer

The polar equation \( r = -1 \) is a circle with radius 1 centered at the origin, expressed as \( x^2 + y^2 = 1 \) in rectangular coordinates.

Step 1: Understanding the Polar Equation

The given polar equation is \( r = -1 \). In polar coordinates, \( r \) is the distance from the origin to the point, and it's negative here, which means the point is in the opposite direction of the positive radial line (line extending directly from the pole outward). This will give us points in the same line but flipped over around the origin.

Step 2: Sketching the Polar Graph

For \( r = -1 \), since the radius is constant at -1, all points lie on the circle centered at the origin with a radius of 1. However, because \( r \) is negative, we plot the points on the opposite side of the pole. So, in the polar graph, it will look like points on the circumference of a circle at a distance of 1 from the pole, but flipped over due to negation.

Step 3: Converting Polar to Rectangular Coordinates

The relationship between polar and rectangular coordinates is given by: \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Substituting \( r = -1 \) into these formulas, we have: \( x = -\cos(\theta) \) and \( y = -\sin(\theta) \).

Step 4: Expressing Equation in Rectangular Form

To express \( r = -1 \) in rectangular coordinates, use the conversion formulas. With \( r^2 = x^2 + y^2 \) and using \( r = -1 \), we have \( (-1)^2 = x^2 + y^2 \), resulting in the equation: \( x^2 + y^2 = 1 \). This is the equation of a circle with radius 1 centered at the origin, which matches our understanding from the polar sketch.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to pinpoint points in the plane with a pair of numerical coordinates. These coordinates are usually denoted as \((x, y)\), where:

• \(x\) represents the horizontal distance from the origin (the point (0,0))
• \(y\) represents the vertical distance from the origin

In a rectangular coordinate system, any point is uniquely determined by these two numbers.
It is essential to understand this system because it lays the groundwork for converting between other coordinate systems like polar coordinates. Through various transformations, equations like \(x^2 + y^2 = 1\), which represent shapes such as circles, can be easily plotted and comprehended.

Polar to Rectangular Conversion

The process of converting polar coordinates to rectangular coordinates helps in utilizing the x-y grid to understand polar equations better.
The conversion is executed through these formulas:

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

Here, \(r\) is the radial distance from the origin, and \(\theta\) is the angular coordinate or angle in radians.
For instance, for \(r = -1\), substituting in the formulas gives us \(x = -\cos(\theta)\) and \(y = -\sin(\theta)\).
Using these, the points derived from polar coordinates are correctly and accurately plotted onto a rectangular coordinate system, such as converting the polar equation \(r = -1\) into the circle \(x^2 + y^2 = 1\).

Equation of a Circle

An equation of a circle in a plane, typically in its standard form, is expressed as \(x^2 + y^2 = r^2\), where \(r\) represents the circle's radius.
This formula arises from the Pythagorean theorem and represents all the points that are a distance \(r\) from a center point, in this case, the origin (0,0).

• It effectively describes a closed, perfectly round shape.
• For the equation \(x^2 + y^2 = 1\), it indicates a circle centered at the origin with a radius of 1.

Whenever a problem involves curves like a circle, especially in contexts of intricate transformations or coordinate conversion tasks, spotting the pattern of this equation is crucial for sketching and solving related challenges accurately.

Graphing Polar Equations

Graphing polar equations focuses keenly on plotting points based on radial distance and angular measurement.

• In polar coordinates, each point on the plane is defined by \((r, \theta)\).
• Here, \(r\) is the distance from the pole (origin), and \(\theta\) is the angle from the polar axis (positive x-axis equivalent).

For instance, plotting \(r = -1\) situates all points at a consistent distance of 1 from the pole, but in the direction opposite to the angle \(\theta\).
These points lay on a circle, embracing a method of graph interpretation distinct from the rectilinear plot.
Switching between plotting styles elevates understanding, as it reveals the different expressions of geometric shapes like circles across distinct coordinate structures. Understanding these graphs aids significantly in visualizing complex mathematical relationships.