Problem 30 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \sin \theta=r \cos \theta $$

Step-by-Step Solution

Answer

The polar equation converts to the Cartesian equation \( y = x \), which is a line through the origin with a slope of 1.

Step 1: Write Down Initial Equation

The given polar equation is \( r \sin \theta = r \cos \theta \).

Step 2: Eliminate r from Both Sides

Since both sides have \( r \), you can divide both sides by \( r \), assuming \( r eq 0 \), to simplify the equation: \( \sin \theta = \cos \theta \).

Step 3: Convert Trigonometric Expression

The equation \( \sin \theta = \cos \theta \) can be rewritten as \( \tan \theta = 1 \).

Step 4: Convert to Cartesian Form

In Cartesian coordinates, the relation \( \tan \theta = 1 \) is represented as \( \frac{y}{x} = 1 \). Cross-multiplying gives \( y = x \).

Step 5: Identify the Graph

The Cartesian equation \( y = x \) represents a straight line through the origin with a slope of 1. This line forms a 45-degree angle with the x-axis.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential in simplifying and solving equations that involve angles. In our initial polar equation, \( r \sin \theta = r \cos \theta \), the trick is recognizing that both sides of the equation can be reduced using trigonometric identities. By dividing both sides by \( r \), assuming \( r eq 0 \), we get \( \sin \theta = \cos \theta \). The significance of this equation lies in the fact that it can be rewritten using another trigonometric identity: \( \tan \theta = 1 \). This identity tells us that the angle \( \theta \) is such that the tangent has a value of 1, which is a key step in our conversion process.
Understanding trigonometric identities can greatly simplify many mathematical problems by transforming terms and expressions into more recognizable or useful forms. In this case, it allowed us to simplify the polar expression into a more straightforward trigonometric equation.

Graph Identification

Graph identification is the process of interpreting mathematical equations to determine their graphical representation. After we converted the polar equation \( r \sin \theta = r \cos \theta \) into the Cartesian form \( y = x \), it becomes much easier to visualize the graph. The equation \( y = x \) represents a straight line passing through the origin. It has a slope of 1, making it a perfectly diagonal line that forms a 45-degree angle with both the x-axis and y-axis.
Graphically, when you see the equation \( y = x \), it tells you about the path or direction this line takes in the Cartesian plane. Graph identification helps in quickly understanding the nature of solutions represented by equations, which becomes crucial, especially in planning and modeling real-world scenarios.

Cartesian Equations

Cartesian equations are used to describe lines, curves, and surfaces in a 2-dimensional or 3-dimensional plane. Unlike polar equations, which deal with circles and angles, Cartesian coordinates express locations in terms of x (horizontal) and y (vertical) coordinates. In our exercise, we convert the polar equation to Cartesian by moving from the trigonometric equation \( \tan \theta = 1 \) to the form \( \frac{y}{x} = 1 \).

• By cross-multiplying, we obtain \( y = x \), representing a straight line.
• This highlights one of the Cartesian equation's advantages: they typically offer a direct and simple interpretation of the relationship between variables.

Therefore, understanding Cartesian equations and their conversions is crucial when translating and interpreting information from one coordinate system to another. This skill is incredibly valuable in fields such as physics, engineering, and computer graphics where design and analysis often demand such conversions.