Problem 58 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ x^{2}+x y+y^{2}=1 $$

Step-by-Step Solution

Answer

The polar equation is \(r^2 = \frac{1}{1 + \cos\theta\sin\theta}\).

Step 1: Understanding Cartesian Coordinates

The given equation is in Cartesian form: \(x^2 + xy + y^2 = 1\). Our task is to convert this equation into polar coordinates using the relationships between Cartesian and polar coordinates.

Step 2: Define Polar Coordinate Relationships

In polar coordinates, any point in the plane can be represented as \((r, \theta)\) where \(x = r\cos\theta\) and \(y = r\sin\theta\). Additionally, \(r^2 = x^2 + y^2\). We'll use these to rewrite the given Cartesian equation.

Step 3: Substitute Polar Relationships

Start by substituting \(x = r\cos\theta\) and \(y = r\sin\theta\) into the equation: \(x^2 + xy + y^2 = (r\cos\theta)^2 + (r\cos\theta)(r\sin\theta) + (r\sin\theta)^2 = 1\).

Step 4: Simplify the Equation

Simplify the expression: \((r\cos\theta)^2 + (r^2 \cos\theta\sin\theta) + (r\sin\theta)^2\). This becomes \(r^2 \cos^2\theta + r^2 \cos\theta\sin\theta + r^2 \sin^2\theta\).

Step 5: Factor and Use Identity

Notice that \(r^2 = r^2\cos^2\theta + r^2\sin^2\theta\) since this follows from the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\). Factor \(r^2\) out: \(r^2(\cos^2\theta + \cos\theta\sin\theta + \sin^2\theta) = 1\).

Step 6: Further Simplify

By observing the terms, \(\cos^2\theta + \sin^2\theta = 1\), so the expression simplifies to \(r^2 (1 + \cos\theta\sin\theta) = 1\).

Step 7: Final Polar Equation

Since we have \(r^2 (1 + \cos\theta\sin\theta) = 1\), we rearrange it to get the polar equation: \(r^2 = \frac{1}{1 + \cos\theta\sin\theta}\).

Key Concepts

Convert Cartesian to Polar

Converting a Cartesian equation into a polar equation involves using the relationship between the two coordinate systems. In Cartesian coordinates, points are defined by their horizontal and vertical distances, represented by \(x\) and \(y\). In polar coordinates, each point is defined by its distance from the origin, \(r\), and the angle, \(\theta\), from the positive \(x\)-axis. This relationship can be described with these conversions:

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

When working with equations like \(x^2 + xy + y^2 = 1\), we substitute \(x\) and \(y\) with \(r\cos\theta\) and \(r\sin\theta\) respectively. This allows rewriting the equation and investigating the pattern that emerges in polar form. Always remember that simplifying is key. For our example, this eventually gives us \(r^2 = \frac{1}{1 + \cos\theta\sin\theta}\).

Polar Coordinates

Polar coordinates are a useful alternative to Cartesian coordinates, especially when dealing with problems involving curves, circles, and angles. Instead of defining a point by \(x\) and \(y\) positions, polar coordinates make use of:

• \(r\) - the radial distance: the straight-line distance from the origin to the point.
• \(\theta\) - the angular coordinate: the angle measured in counterclockwise direction from the positive \(x\)-axis.

This system simplifies many mathematical operations, especially those involving rotational symmetry or central points. For someone transitioning from Cartesian coordinates, it may require practice, but the ability to analyze problems in polar space can offer remarkable insights and simplifications for circular and angular situations. Polar coordinates can thus help solve real-world problems efficiently, especially in fields like physics and engineering.

Trigonometric Identities

Trigonometric identities play a vital role in simplifying both Cartesian and polar equations. These identities are fundamental relationships involving trigonometric functions like sine and cosine. One of the most common and useful identities is the Pythagorean identity:

• \(\cos^2\theta + \sin^2\theta = 1\)

This identity helps in consolidating expressions that might initially seem complicated. For example, when converting from Cartesian to polar coordinates, this identity helps simplify terms like \(r^2(\cos^2\theta + \sin^2\theta)\) to simply \(r^2\). Understanding and using these identities are crucial when dealing with transformations between coordinate systems, ensuring that simplification leads to more manageable equations, as demonstrated with the final form \(r^2 = \frac{1}{1 + \cos\theta\sin\theta}\). Mastering these identities is a stepping stone towards efficiently handling diverse mathematical problems.