Problem 51 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ x=y $$

Step-by-Step Solution

Answer

The equivalent polar equation is \(\theta = \frac{\pi}{4} + n\pi\).

Step 1: Understand the Relationship Between Cartesian and Polar Coordinates

Cartesian coordinates are expressed as \(x, y\), while polar coordinates use \(r, \theta\). The relationships between them are \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Step 2: Substitute Cartesian Equations with Polar Equations

Since the given equation is \(x = y\), substitute the Cartesian relationships with polar coordinates: \(r\cos(\theta) = r\sin(\theta)\).

Step 3: Simplify the Polar Equation

To simplify \(r\cos(\theta) = r\sin(\theta)\), divide both sides by \(r\) (given that \(r eq 0\)). This yields: \(\cos(\theta) = \sin(\theta)\).

Step 4: Determine the Equation Form

Simplify \(\cos(\theta) = \sin(\theta)\) using trigonometric identities: \(\cos(\theta) = \sin(\theta)\) can be rewritten as \(\frac{\cos(\theta)}{\sin(\theta)} = 1\). This leads to \(\cot(\theta) = 1\), indicating \(\theta = \frac{\pi}{4} + n\pi\), where \(n\) is an integer.

Key Concepts

Cartesian Coordinates

The Cartesian coordinate system is a two-dimensional plane where each point is determined by a pair of numerical values, typically referred to as \( (x, y) \). This system is named after the French mathematician René Descartes. It divides the plane into four quadrants, defined by the intersection of the \( x \)-axis (horizontal) and the \( y \)-axis (vertical).
Each point on the plane is specified in terms of its distance from these axes, allowing for precise localization of points. For example, the point \( (2, 3) \) represents a spot 2 units horizontally and 3 units vertically from the origin \( (0, 0) \).
In problems involving coordinate transformation to polar coordinates, this familiar structure is often the starting point. Cartesian coordinates play a crucial role in visualizing mathematical equations, making them relatable to real-life measurements like distance and direction.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are very useful in simplifying expressions and are central to solving geometric and algebraic problems.
One important pair of identities used in transforming Cartesian equations to polar equations is:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( an(\theta) = \frac{y}{x} \)

These relationships link the two coordinate systems together, enabling conversion between them. For example, in the equation \( \cos(\theta) = \sin(\theta) \), division leads us to conclude that \( \cot(\theta) = 1 \), indicating specific angles that can be calculated.
Understanding these identities allows one to find angles and lengths within polygons and circles, making them particularly useful in calculations involving waves, oscillations, and circles.

Coordinate Transformation

Coordinate transformation is the process of converting points from one coordinate system to another. In the context of this exercise, we are transforming Cartesian coordinates into polar coordinates.
The polar coordinate system describes each point by its distance \( r \) from the origin and an angle \( \theta \) from the reference direction, which is often the positive \( x \)-axis. This system is particularly useful for dealing with problems involving circular and rotational symmetry.
When transforming from Cartesian to polar, one employs the following formulas:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(\frac{y}{x}) \)

For instance, the Cartesian equation \( x = y \) translates into \( \cos(\theta) = \sin(\theta) \), which simplifies to \( \theta = \frac{\pi}{4} + n\pi \), providing an elegant representation in polar form.
Mastering coordinate transformation not only simplifies complex algebraic problems but also provides insights into a wide range of scientific and engineering applications.