Problem 58 - College Algebra

Question

Express each equation in polar form with \(r\) as a function of \(\theta\). $$ 2 x^{2}+4 x y+2 y^{2}=9 $$

Step-by-Step Solution

Answer

In polar form, \(r = \sqrt{\frac{9}{2(1+\sin 2\theta)}}\)."

Step 1: Recognize the Cartesian equation

The given equation in Cartesian coordinates is \(2x^2 + 4xy + 2y^2 = 9\).

Step 2: Identify polar coordinate expressions

In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 3: Substitute polar coordinates into the equation

Replace \(x\) and \(y\) in the equation with polar expressions: \(2(r \cos \theta)^2 + 4(r \cos \theta)(r \sin \theta) + 2(r \sin \theta)^2 = 9\).

Step 4: Simplify the expression

Simplify the expression: \(2r^2 \cos^2 \theta + 4r^2 \cos \theta \sin \theta + 2r^2 \sin^2 \theta = 9\).

Step 5: Factor out \(r^2\)

Factor out \(r^2\) from the left-hand side: \(r^2 (2 \cos^2 \theta + 4 \cos \theta \sin \theta + 2 \sin^2 \theta) = 9\).

Step 6: Simplify using trigonometric identities

Use \(\cos^2 \theta + \sin^2 \theta = 1\) to simplify: \(r^2 (2 + 4 \cos \theta \sin \theta) = 9\).

Step 7: Solve for \(r\) in terms of \(\theta\)

Since the equation remains complex, use additional identities or approaches to express \(r\) as a function of \(\theta\): \(r^2 (2 + 2 \sin 2\theta) = 9\).

Step 8: Divide and simplify the solution

Solve for \(r\): \(r^2 = \frac{9}{2(1+\sin 2\theta)}\), and therefore, \(r = \sqrt{\frac{9}{2(1+\sin 2\theta)}}\).

Key Concepts

Cartesian coordinates

Cartesian coordinates are a mathematical system for representing points on a plane using two numbers, typically referred to as the x-coordinate and the y-coordinate. This is a very intuitive way to describe positions in a two-dimensional space, as it aligns with the usual way we view the world, with horizontal (x-axis) and vertical (y-axis) directions.

In this exercise, the equation given is \[ 2x^2 + 4xy + 2y^2 = 9 \].

Here, both x and y are examples of Cartesian coordinate variables. We use these coordinates because they provide a straightforward method to express locations and equations in a flat, rectangular coordinate system.

• The x-coordinate corresponds to horizontal movement.
• The y-coordinate corresponds to vertical movement.

By transforming equations in this form, we can explore different coordinate systems that can offer deeper insights or simplify complex equations.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are always true for any angle. These identities play a pivotal role in manipulating and simplifying expressions involving trigonometric functions, especially when converting equations between different coordinate systems.

A key identity used in this exercise is:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)

This identity helps simplify polar coordinates expressions, making it easier to translate from Cartesian forms.

For instance, when simplifying the exercise's expression:\[ 2r^2 \cos^2 \theta + 4r^2 \cos \theta \sin \theta + 2r^2 \sin^2 \theta \]we apply the identity, combining terms to reduce complexity.

Furthermore, another identity, \( \sin 2\theta = 2 \sin \theta \cos \theta \), helps in further simplification by combining products of sine and cosine into a single term.

equations in polar form

Equations in polar form represent a fundamental shape or path in a space described by radial distance \(r\) and angle \(\theta\). Unlike Cartesian coordinates, which use x and y to specify locations, polar coordinates focus on the direction and distance from a central point.

In polar form, an equation might look like \(r = f(\theta)\), where \(f(\theta)\) defines how the distance from the origin changes as the angle varies.

The task in this problem is to express a Cartesian equation in a polar coordinate form, which helps in:

• Visualizing equations related to circles, spirals, and other curves more naturally.
• Solving complex problems involving rotational symmetry or radial movements effectively.

For our given problem, the Cartesian form,\[ 2x^2 + 4xy + 2y^2 = 9 \], when translated to polar form becomes \[ r = \sqrt{ \frac{9}{2(1 + \sin 2\theta)}} \].

This step involves replacing Cartesian variables with polar counterparts (\(x = r \cos \theta, y = r \sin \theta\)). Then using identities to simplify.

conversion between coordinate systems

Conversion between coordinate systems involves translating points or equations from one form to another, such as from Cartesian to polar or vice versa. This is essential when a specific representation provides a clearer understanding of the problem or makes calculations easier.

To convert from Cartesian to polar, the relationships are:

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

The exercise is a classic example where we start with a Cartesian equation and use these relationships to replace x and y variables with their polar equivalents.

By substituting these expectations, the complex relationships in Cartesian coordinates can often become simpler to understand and work within polar form, especially when dealing with symmetrical or rotational problems. This technique is widely used in physics, engineering, and computer graphics for its ability to simplify complex coordinate transformations.