Problem 68 - Algebra and Trigonometry

Question

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$ r=\cos \theta $$

Step-by-Step Solution

Answer

The rectangular form of the polar equation \( r=\cos \theta \) is \( x=1 \). The graph for this equation is a vertical line through \( x=1 \).

Step 1: Conversion to Rectangular Form

The rectangular coordinates \( x \) and \( y \) can be written in terms of the polar coordinates \( r \) and \( \theta \) as \( x = r \cos \theta \) and \( y = r \sin \theta \). As from the given polar equation \( r = \cos \theta \), substituting \( r \) into \( x = r \cos \theta \) gives us the rectangular form. So, the rectangular form is \( x = r = \cos \theta \).

Step 2: Graphing the Rectangular Equation

The rectangular equation \( x = \cos \theta \) represents a vertical line passing through \( x=1 \) on a Cartesian coordinate system. This is because \( \cos \theta \) ranges between -1 and 1, but in the polar coordinate system, \( r = \cos \theta \) only takes on values between 0 and 1, due to the distance \( r \) being always non-negative. Thus, the graph will be a line segment from (1, 0) upward.

Key Concepts

Polar Equation Conversion

When working with polar equations, one of the critical skills is converting these equations into rectangular form for easier analysis and graphing. To convert a polar equation like \( r = \cos \theta \) to rectangular coordinates, we use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). These relationships arise from the definition of cosine and sine in the context of right-angled triangles. To illustrate, in the given exercise, setting \( r \) equal to \( \cos \theta \) allows us to directly replace \( r \) in the related trigonometric expression to find that \( x = \cos \theta \), and in this particular case, the equation simplifies so that it does not explicitly involve \( y \), indicating that any value of \( y \) is acceptable, as long as \( x \) is equal to \( \cos \theta \).

Understanding these conversions is essential because it lets students shift perspectives between polar and rectangular systems, often making it easier to solve problems, analyze functions, or understand the geometry of curves sophisticatedly.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane defined by a pair of perpendicular axes: typically, the horizontal axis (x-axis) and the vertical axis (y-axis). Points in this system are defined by an ordered pair of numbers \( (x, y) \), representing horizontal and vertical distances from the system's origin.

In contrast with polar coordinates that are given as \( (r, \theta) \), with \( r \) being the radial distance from the origin and \( \theta \) being the angle from the positive x-axis, the rectangular coordinate system is much more intuitive for visualizing linear equations, as each axis corresponds directly to one of the variables in the equation. The system is essential for graphing equations and understanding geometrical shapes in a plane. It's the most commonly used coordinate system for practical applications in science and engineering.

Graphing Polar Equations

Graphing polar equations involves plotting points in the polar coordinate system and then connecting these points to visualize the curve they form. However, the polar coordinate system isn't always the most straightforward way to understand these curves, which is why converting and graphing on a rectangular system can be helpful.

As described in the original exercise, once the polar equation \( r = \cos \theta \) is converted to its rectangular form \( x = \cos \theta \) by substitution, graphing in the rectangular system becomes much easier. The resulting equation corresponds to a vertical line since \( x \) is constant and \( y \) can take any value. This equation does not yield the typical circle or spiral that you may expect from polar equations, because the conversion has simplified the expression to a form that represents only a portion of the polar graph, specifically where the radial distance \( r \) is non-negative.

Trigonometry in Cartesian Plane

Trigonometry in the Cartesian plane is an extension of trigonometry from the unit circle to the entire plane. The basic trigonometric functions, sine and cosine, originally defined for angle measures, correspond to the x and y coordinates of a point on the unit circle. On the Cartesian plane, these functions are used to relate the polar coordinates \( r \) and \( \theta \) to the rectangular coordinates \( x \) and \( y \).

For example, the cosine function relates the angle \( \theta \) to the x-coordinate of a point, and the sine function relates \( \theta \) to the y-coordinate, as seen in the equations \( x = r \cos \theta \) and \( y = r \sin \theta \). This relationship is essential in converting between polar and rectangular forms. Trigonometric identities also play a crucial role in simplifying equations or in more complex topics such as Fourier analysis, which crosses over into various fields of study including physics, engineering, and signal processing.