Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the Cartesian equations of the graphs of the given polar equations. $$ r \sin \theta-1=0 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( y = 1 \).

Step 1: Understanding the Polar Equation

The given polar equation is \( r \sin \theta - 1 = 0 \). This means \( r \sin \theta = 1 \). In polar coordinates, \( r \) represents the radius, and \( \theta \) represents the angle in radians from the positive x-axis.

Step 2: Relation to Cartesian Coordinates

In Cartesian coordinates, the relationship to polar coordinates is \( x = r \cos \theta \) and \( y = r \sin \theta \). Since \( r \sin \theta = y \) in Cartesian form, we directly use this relation.

Step 3: Solve for Cartesian Equation

Since we have \( r \sin \theta = 1 \), using the relation \( y = r \sin \theta \), we set \( y = 1 \). This is already a Cartesian equation. Therefore, the polar equation \( r \sin \theta - 1 = 0 \) translates to the Cartesian equation \( y = 1 \).

Key Concepts

Polar Equations

Polar equations are a way to describe curves using a relationship between the distance from a fixed point (called the pole or origin) and an angle. Instead of the usual x and y coordinates we use in Cartesian equations, polar equations make use of a radial coordinate, denoted as \( r \), and an angular coordinate, \( \theta \).

• \( r \) represents the radius or the distance from the origin.
• \( \theta \) represents the angle measured from the positive x-axis.

Polar equations are particularly useful for describing shapes that have clear radial symmetry, such as circles and spirals. A typical polar equation like \( r = f(\theta) \) directly relates these polar coordinates. In our example, we have \( r \sin \theta = 1 \), which forms a horizontal line in Cartesian coordinates when transformed, illustrating how polar equations can represent curves in a different yet insightful manner.

Coordinate Transformation

Coordinate transformation involves converting points or equations from one coordinate system to another, which is highly useful in geometry and physics. When we transform the given polar equation \( r \sin \theta = 1 \) to Cartesian coordinates, we use the relationships:

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

This transformation process helps us switch from the circular-based polar system to the grid-based Cartesian system. For the given exercise, since \( r \sin \theta = y \), we substitute directly to get \( y = 1 \). Essentially, we've moved from describing a certain curve as a function of radial distance and angle, to describing it with plain horizontal and vertical distances. This is helpful because it can make plotting and interpreting the curves on standard graph paper much easier.

Trigonometric Functions

Trigonometric functions are the tools we use to relate angles to ratios of the sides of right triangles. They are fundamental in the transformation between polar and Cartesian coordinates. For instance, the sine function, which is crucial in our example, equates to the opposite side over the hypotenuse in a right triangle.

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• In the polar context, \( r \sin \theta \) directly gives us the y-coordinate in Cartesian form.

For the specific equation \( r \sin \theta = 1 \), it states that the y-value is constantly 1, no matter the value of \( \theta \) assuming \( r \) and \( \theta \) are such that they satisfy the equation. This invariance shows that trigonometric functions not only help in describing oscillations and cycles but also in understanding geometric transformations.