Problem 91 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ r=-\sqrt{5} \csc (\theta) $$

Step-by-Step Solution

Answer

The equation in rectangular coordinates is \( y = -\sqrt{5} \).

Step 1: Rewrite Cosecant in Terms of Sine

We know that cosecant is the reciprocal of sine, so we can rewrite the equation as \( r = -\frac{\sqrt{5}}{\sin(\theta)} \).

Step 2: Use the Identity for \(r\sin(\theta)\)

Recall the polar to rectangular coordinate conversion identities: \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Here, multiply both sides of the given equation by \( \sin(\theta) \) to get \( r\sin(\theta) = -\sqrt{5} \).

Step 3: Substitute into Rectangular Equation

Since \( y = r\sin(\theta) \), substitute \( y \) for \( r\sin(\theta) \) in the equation to obtain \( y = -\sqrt{5} \).

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are fundamental in simplifying and converting equations, especially when switching between different coordinate systems. For this particular exercise, we focused on the cosecant function. Cosecant, denoted as \( \csc(\theta) \), is the reciprocal of the sine function, \( \sin(\theta) \). Thus, \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Knowing this identity allows us to rewrite equations and make them easier to manipulate.

This transformation is a crucial step during conversions. In our equation, by expressing \( \csc(\theta) \) in terms of \( \sin(\theta) \), we obtained \( r = -\frac{\sqrt{5}}{\sin(\theta)} \). Remember, the key to mastering trigonometric identities is practice and familiarity with how they interrelate.

Exploring Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use a two-dimensional plane represented by \( (x, y) \). The horizontal axis is called the x-axis and the vertical is the y-axis. This coordinate system is ubiquitous in mathematics because it allows for precise representation and analysis of shapes and equations.

In the step-by-step solution, we transitioned from polar to rectangular coordinates using the identities \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). This is how coordinates are transformed from the circular path of polar coordinates into the linear path of rectangular ones. In the given problem, we focused on \( y \), which represents the vertical position on the Cartesian plane. We derived an equation where \( y = -\sqrt{5} \), indicating a straight horizontal line at \( -\sqrt{5} \) on the y-axis.

Understanding Polar Coordinates

Polar coordinates offer a different way of looking at a point's position using a radius and angle, \( (r, \theta) \), instead of the linear \( (x, y) \). In this system, \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis. This system excels in scenarios involving circular or rotational symmetry.

Through this exercise, we learned that translating from polar to rectangular coordinates relies heavily on trigonometric functions. With polar coordinates \( r = -\sqrt{5} \csc(\theta) \), the main task was transforming this into information usable in a rectangular plane. By replacing the polar expressions with their rectangular equivalents, such as \( r\sin(\theta) = y \), we draw on a powerful method for visuospatial understanding of equations. Thus, understanding polar coordinates enriches your mathematical toolset by allowing you to decipher complex problems with ease.