Problem 22 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=\cos \theta $$

Step-by-Step Solution

Answer

The polar equation \( r = \cos \theta \) transforms to \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\), a circle in rectangular coordinates.

Step 1: Understand the Polar Equation

The given polar equation is \( r = \cos \theta \). This equation describes the relationship between the radius \( r \) and the angle \( \theta \) in polar coordinates.

Step 2: Transform the Equation to Rectangular Coordinates

In rectangular coordinates, we can use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). We also know that \( r^2 = x^2 + y^2 \).Using the equation \( r = \cos \theta \), substitute \( r = \frac{x}{r} \) in polar coordinates:\[ r = \frac{x}{r} \rightarrow r^2 = x \rightarrow x^2 + y^2 = x \].Thus, the rectangular form is \( x^2 + y^2 = x \).

Step 3: Rearrange Rectangular Equation for Graphing

To sketch the graph, rearrange \( x^2 + y^2 = x \) to \( x^2 - x + y^2 = 0 \). Complete the square for the terms involving \( x \):\[ x^2 - x = (x - \frac{1}{2})^2 - \frac{1}{4} \].Thus, the equation becomes \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\), which is a circle centered at \( (\frac{1}{2}, 0) \) with radius \( \frac{1}{2} \).

Step 4: Sketch the Polar Graph

In polar coordinates, \( r = \cos \theta \) represents a circle centered at \( (\frac{1}{2}, 0) \) with radius \( \frac{1}{2} \). As \( \theta \) varies from 0 to \( 2\pi \), \( r \) varies accordingly, completing a circle. Plot this on polar graph paper with the circle passing through points \((1, 0^\circ)\), \((0, 90^\circ)\), \((-1, 180^\circ)\), and \((0, 270^\circ)\).

Key Concepts

Rectangular Coordinates

When working with coordinates, you often encounter two primary systems: polar and rectangular. Rectangular coordinates are more commonly known as Cartesian coordinates, and they are expressed as \((x, y)\), where:

• \(x\) is the distance from the point to the vertical y-axis, measured horizontally.
• \(y\) is the distance from the point to the horizontal x-axis, measured vertically.

Rectangular coordinates form a grid of horizontal and vertical lines that we use to locate points in a plane.
This system is especially useful for describing locations on a straight grid, making calculation and visualization intuitive and straightforward. The formula to relate polar coordinates \((r, \theta)\) to rectangular coordinates is:

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

This transformation allows us to convert data between these two systems easily, beneficial for many applications such as computer graphics, engineering, and physics.

Polar Equations

Polar equations describe relationships using polar coordinates. These are expressed in terms of \(r\) (the radius or the distance from the origin) and \(\theta\) (the angle with the positive x-axis). For instance, the equation \(r = \cos \theta\) implies that the radius changes based on the cosine of the angle.
Polar coordinates are particularly advantageous when dealing with curves and shapes that have radial symmetry, such as circles or spirals. They simplify equations that might be more complex in rectangular coordinates.

• Polar equations can describe a variety of figures, including circles, spirals, and rose curves.
• They offer a concise way to represent curves that are symmetric around a point, such as those found in waves or oscillatory patterns.

In the case of \(r = \cos \theta\), this is a polar equation representing a circle in the polar coordinate plane, centered on the polar axis (x-axis) at \( (\frac{1}{2}, 0) \), with a radius of \(\frac{1}{2}\).

Coordinate Transformation

Coordinate transformation is the process of converting coordinates from one system to another, such as from polar to rectangular coordinates. This is vital because different problems are more easily solved in different systems.
When transforming polar to rectangular coordinates, we use the relationships:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r^2 = x^2 + y^2\)

For example, to convert the polar equation \(r = \cos \theta\) into rectangular form, we replace \(r\) with \(\sqrt{x^2 + y^2}\) to obtain \(r^2 = x\). Then, transforming gives \(x^2 + y^2 = x\), which, when rearranged, describes a circle.
This transformation allows us to see how equations representing shapes in polar can be viewed through the lens of rectangular coordinates, which can help in understanding the implications or effects of the equation in a different context or coordinate system.