Problem 65 - Algebra and Trigonometry

Question

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$ r=4 \csc \theta $$

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r = 4\csc\theta\) is \(y = 4\), which represents a horizontal line 4 units above the x-axis.

Step 1: Convert Polar Equation to Rectangular Equation

For any point P in the Cartesian plane, let \(r\) and \(\theta\) denote the polar coordinates of P, and let (x, y) denote the rectangular coordinates of P. The relationships between polar and Cartesian coordinates are given by: \(r = \sqrt{x^2+y^2}\) and \(\theta = \arctan(\frac{y}{x})\). We start by using the trigonometric identity \(\csc\theta = \frac{1}{\sin\theta}\) to rewrite the given polar equation \(r = 4\csc\theta\) as \(r = \frac{4}{\sin\theta}\). Then multiply each side by \(\sin\theta\), we have \(r\sin\theta = 4\). In terms of cartesian coordinates, \(r\sin\theta = y\). Therefore, the rectangular form of the polar equation is \(y = 4\).

Step 2: Graph the Rectangular Equation

The equation \(y = 4\) describes a horizontal line that is 4 units above the x-axis. Starting at the point (0, 4) on the y-axis, draw a straight line going to the right and to the left.

Key Concepts

Polar Coordinates

Understanding polar coordinates is essential for tackling problems related to different coordinate systems. In polar coordinates, each point on a plane is defined by two values: the radial distance from a central point called the origin, and an angle measured from a reference direction, usually the positive x-axis. Specifically, the radial distance is denoted by the symbol r, and the angle is represented by the Greek letter θ (theta).

Polar coordinates are particularly useful in situations where relationships are more naturally expressed in terms of angles and distances from a central point. They're commonly seen in fields like physics, particularly when dealing with circular or spiral paths and when analyzing wave patterns. Converting polar equations to rectangular coordinates can be crucial in understanding the shape and position of curves relative to a more familiar Cartesian system.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are what most people visualize when they think of a coordinate plane. This system uses two perpendicular axes: the horizontal axis is known as the x-axis, and the vertical axis is the y-axis. The location of a point in this plane is given by an ordered pair (x, y), where x and y represent the distances from the y-axis and x-axis, respectively.

Switching from polar to rectangular coordinates involves using trigonometric identities and relationships to rewrite radial distances and angles as x and y coordinates. Learning to convert between coordinate systems is a valuable skill, as it allows students to solve complex geometrical and physical problems by choosing the most suitable framework for the given conditions.

Trigonometric Identities

Trigonometric identities are essentially equations that are true for all values of the variables involved. They are the backbone of solving numerous mathematical problems, especially when it comes to converting between polar and rectangular coordinate systems. Identities like sin²θ + cos²θ = 1 and tanθ = sinθ/cosθ are fundamental in such conversions.

When we have expressions involving trigonometric functions in polar coordinates, such as 4 csc θ, we can use identities like cscθ = 1/sinθ to rewrite them in terms of sine and cosine. These identities allow us to express radial distances and angles as a relationship between x and y, thus transitioning smoothly from polar to rectangular equations. For students, mastering these identities translates to greater ease in moving between coordinate systems, which is a critical skill in various mathematical and engineering applications.