Problem 29 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x y=-3$$

Step-by-Step Solution

Answer

The polar equation is \( r^2 \cos \theta \sin \theta = -3 \).

Step 1: Understand the rectangular equation

We start with the equation given in rectangular coordinates, which is \[ x y = -3. \] This equation represents a curve in the Cartesian plane.

Step 2: Use polar coordinates conversion formulas

Recall the conversions between rectangular and polar coordinates: \[ x = r \cos \theta \] \[ y = r \sin \theta. \] We will substitute these expressions into the original equation.

Step 3: Substitute polar equivalents into the equation

Replace \(x\) and \(y\) in the equation \( xy = -3 \) with their polar equivalents. The equation becomes: \[ (r \cos \theta)(r \sin \theta) = -3. \]

Step 4: Simplify the polar equation

Simplify the equation obtained by substitution: \[ r^2 \cos \theta \sin \theta = -3. \] This is the polar form of the original Cartesian equation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to plot points on a plane using two numbers: the x-coordinate and the y-coordinate. These coordinates are named after the mathematician and philosopher René Descartes. To visualize it, imagine a graph with two intersecting straight lines. The horizontal line is the x-axis, and the vertical line is the y-axis. Together, they form the Cartesian plane.

In this plane:

• The x-coordinate tells you how far to move horizontally from the origin (where the axes intersect).
• The y-coordinate tells you how far to move vertically.
• Each point in the plane can be represented as an ordered pair (x, y).

These coordinates are fundamental in algebra and calculus, allowing us to plot graphs and see the shape of equations in visual form.

Cartesian Plane

The Cartesian plane is a two-dimensional surface created by the intersection of two perpendicular lines, the x-axis, and the y-axis. It's like laying graph paper flat on a desk, where each meeting of lines represents a unique point.

This plane is divided into four quadrants:

• Quadrant I (top-right) where both x and y are positive.
• Quadrant II (top-left) where x is negative and y is positive.
• Quadrant III (bottom-left) where both x and y are negative.
• Quadrant IV (bottom-right) where x is positive and y is negative.

Understanding these quadrants is crucial when dealing with functions that change sign based on their position on the plane. By recognizing where each point lies, we can better predict and describe the behavior of mathematical formulas in real-world contexts.

Polar Coordinates Conversion

Converting between rectangular and polar coordinates is a common task in mathematics, especially when dealing with curves and circular patterns. The two systems aim to locate points but do so differently. Polar coordinates use a radius and an angle, denoted as (r, θ), where:

• r represents the distance from the origin to the point.
• θ represents the angle between the positive x-axis and the point.

To convert from rectangular coordinates (x, y) to polar coordinates (r, θ):

• Use the formula \( r = \sqrt{x^2 + y^2} \) to find the distance.
• Determine the angle with \( \theta = \tan^{-1}(y/x) \).

Conversely, to convert from polar to rectangular, use:

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

These conversions allow us to express certain equations more naturally in polar form, especially when dealing with circular or sinusoidal shapes.

Curves in Mathematics

Curves in mathematics describe smooth, continuous lines or the path traced by a point as it moves through space according to defined mathematical rules. These can range from simple lines and circles to complex and intricate shapes seen in calculus and geometry.

Curves express diverse phenomena:

• Lines are straight, with no curvature, representing simple direct equations like y = mx + b.
• Circles, ellipses, and hyperbolas provide insight into conic sections, each forming unique curves with specific properties.
• Parabolas are U-shaped and often depict quadratic equations, following the form \( y = ax^2 + bx + c \).

In advanced mathematics, curves may also need polar equations, like our original example of converting \( xy = -3 \) into a polar form. Understanding curves helps in fields such as engineering, physics, and computer graphics, where the movement through space and the effect of different forces can influence the path of an object.