Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r(3 \cos \theta-4 \sin \theta)=12 $$

Step-by-Step Solution

Answer

The Cartesian equation is \(3x - 4y = 12\), a line in the Cartesian plane.

Step 1: Understand the Polar Equation

The given polar equation is \( r(3 \cos \theta - 4 \sin \theta) = 12 \). This equation is in the form involving \(r\), \(\cos \theta\), and \(\sin \theta\), which we can convert to Cartesian coordinates (\(x\), \(y\)).

Step 2: Convert Polar Equation to Cartesian Coordinates

Use the relationships between polar and Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute these into the equation: \( r = \sqrt{x^2 + y^2} \), \( \cos \theta = \frac{x}{r} \), and \( \sin \theta = \frac{y}{r} \).

Step 3: Substitute into the Polar Equation

Replace \(\cos \theta\) and \(\sin \theta\) with \(\frac{x}{r}\) and \(\frac{y}{r}\) respectively in the equation: \[ r(3 \frac{x}{r} - 4 \frac{y}{r}) = 12 \].

Step 4: Simplify the Equation

Simplify the expression: \[ 3x - 4y = 12 \]. This is the Cartesian equation that describes the same graph in the Cartesian plane as the original polar equation does in the polar plane.

Step 5: Sketch Graph in Cartesian Plane

This equation, \(3x - 4y = 12\), represents a straight line in the Cartesian plane. You can sketch the line by finding the x-intercept (\(x = 4\), \(y = 0\)) and y-intercept (\(x = 0\), \(y = -3\)). Connect these points to draw the line.

Key Concepts

Polar Coordinates

Polar coordinates are a way of locating a point in a plane using a distance and an angle. Unlike Cartesian coordinates, which use a grid of horizontal and vertical lines to pinpoint a location, polar coordinates specify the location using:

• An angle, \( \theta \), from a reference direction (typically the positive x-axis).
• A radius, \( r \), which is the distance from the origin to the point.

This system can be especially useful in contexts where angles and distances are more intuitive, such as navigation or circular motion.
In our exercise, the polar equation involves the variables \( r \), \( \cos \theta \), and \( \sin \theta \). Translating this to Cartesian coordinates involves substituting these polar terms with their counterparts in the Cartesian coordinate system. Understanding polar coordinates is crucial because it helps in recognizing patterns and symmetries that aren't obvious in the rectangular coordinate system.

Cartesian Coordinates

Cartesian coordinates use a pair of numbers to specify points on a two-dimensional plane. These coordinates are given by:

• \( x \): The horizontal distance from the origin.
• \( y \): The vertical distance from the origin.

The Cartesian system is highly intuitive since it's based on a grid pattern. This makes it an excellent choice for drawing graphs and plotting equations.
In the polar to Cartesian conversion exercise, the original polar equation \( r(3 \cos \theta - 4 \sin \theta) = 12 \) is transformed into the Cartesian equation \( 3x - 4y = 12 \) using known relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)

This conversion results in an equation of a line, which can be easily plotted using the Cartesian coordinates.

Graphing Equations

Graphing equations is an essential skill in mathematics, enabling us to visualize relationships between variables. By translating equations into graphical forms, we can better understand the solution's behavior.
In Cartesian coordinates, equations are often simpler to interpret visually. Our Cartesian equation \( 3x - 4y = 12 \) represents a line. To graph it:

• Find the x-intercept by setting \( y = 0 \) which gives \( x = 4 \).
• Find the y-intercept by setting \( x = 0 \) which gives \( y = -3 \).

Connect these intercepts to form a straight line, illustrating how changes in \( x \) result in changes in \( y \). This line graph clearly shows the relationship between these variables and allows us to easily interpret the polar equation's Cartesian representation.

Trigonometric Functions

Trigonometric functions are the building blocks of relating angles to side lengths in right triangles, and they frequently appear in equations involving circular or rotational contexts. The three primary trigonometric functions are:\

• \( \sin \theta \): Opposite side to angle \( \theta \) over the hypotenuse.
• \( \cos \theta \): Adjacent side to angle \( \theta \) over the hypotenuse.
• \( \tan \theta \): Opposite side over the adjacent side to \( \theta \).

These functions are critical in converting polar equations to Cartesian forms. In our exercise, \( \cos \theta \) and \( \sin \theta \) were substituted with \( \frac{x}{r} \) and \( \frac{y}{r} \) respectively, based on the geometric interpretations of these ratios.
Trigonometric functions thus allow for the transformation between coordinate systems, making them a cornerstone for understanding polar to Cartesian conversions. Knowing how to apply these functions effectively can be a huge advantage in solving complex equations and understanding wave patterns.