Problem 66 - Precalculus : Building Concepts and Connections

Question

Convert each of the given polar equations to rectangular form. $$r^{2} \cos 2 \theta=1$$

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r^{2} \cos 2 \theta=1\) is \(x^{2} = y^{2}\)

Step 1: Express r in terms of x and y

To convert polar to rectangular form, replace \(r^{2}\) with \(x^{2}+y^{2}\). So our equation becomes \((x^{2}+y^{2}) \cos 2 \theta=1\)

Step 2: Express cos 2 theta in terms of x and y.

Recall that \(\cos\theta= \frac{x}{r}\), with \(r=\sqrt{x^{2}+y^{2}}\). Therefore, \(\cos 2\theta= 2(\frac{x}{(x^{2}+y^{2})^{1/2}})^{2}-1\). Our equation becomes: \((x^{2}+y^{2})(2(\frac{x}{(x^{2}+y^{2})^{1/2}})^{2}-1)=1\)

Step 3: Simplification

Solve the equation for zero and simplify: \(2x^{2} - (x^{2} + y^{2}) = 0\). This finally gives: \(x^{2} = y^{2}\)

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane. They are defined by two values:

• A distance from the origin, also known as the "radius," denoted as \( r \).
• An angle, denoted by \( \theta \), which is measured from the positive x-axis.

This system is particularly useful when dealing with problems involving circles or rotation because it simplifies many mathematics equations. When a point is described in polar coordinates, it means that the point is located \( r \) units from the origin point \( (0,0) \) and \( \theta \) is the direction in which it faces. The beauty of polar coordinates is that they give a unique way of conceptualizing space, especially in scenarios where standard x, y coordinates make the visualization complex. Our task involves translating this understanding into another familiar system: rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, use a grid system defined by two perpendicular axes, x and y. Each point in this system corresponds to an ordered pair \( (x, y) \). Here, \( x \) represents the horizontal distance from the origin, and \( y \) denotes the vertical distance. Rectangular coordinates are intuitive for visualizing horizontal and vertical relationships but can become cumbersome in circular or rotational contexts. That's why we convert polar coordinates into rectangular form for specific applications, using formulas derived from the relationships between the systems, such as \( x = r \cos \theta \) and \( y = r \sin \theta \). Through understanding both systems, we can fluidly transition between perspectives as needed.

Trigonometric Identities

Trigonometric identities are mathematical equations that use sine, cosine, and other trigonometric functions to express relationships between angles and sides of triangles. In the context of converting polar coordinates to rectangular coordinates, these identities become vital tools.

• The basic identities are \( \cos \theta = \frac{x}{r} \) and \( \sin \theta = \frac{y}{r} \).
• Using these, we derive formulas like \( r^2 = x^2 + y^2 \) and relationships for \( \cos 2\theta \) utilizing double angle formulas.
• The identity \( \cos 2\theta = 2\cos^2 \theta - 1 \) helps us transform complex polar equations into cleaner rectangular form equations.

Such identities are tools for simplifying and solving trigonometric equations, making them essential for conversions between polar and rectangular coordinate systems.

Conversions

Conversions between polar and rectangular coordinates rely on understanding both systems' relationship. The conversion process involves the use of specific formulas:

• For converting from polar to rectangular: \( x = r \cos \theta \) and \( y = r \sin \theta \).
• This gives \( r^2 = x^2 + y^2 \), fundamental when our given equation features \( r^2 \cos 2 \theta \).

The idea is to substitute polar expressions with their rectangular equivalents and simplify the resulting equation. For instance, in our original exercise, by substituting these conversions, we simplified \( r^2 \cos 2 \theta = 1 \) to \( x^2 = y^2 \). This new equation can be analyzed using straightforward arithmetic, thus giving a clearer understanding of the relationship between variables.

Mathematics Education

Mathematics education seeks to build foundational understanding through clear instructional strategies and exercises. Conversion problems, like the one we've tackled, help learners bridge gaps between different coordinate systems and see interconnected mathematics frameworks. Such exercises teach:

• Multistep problem-solving approaches.
• How to apply trigonometric identities in practical contexts.
• The importance of understanding underlying concepts rather than just memorizing procedures.

By engaging with conversion problems, students develop logical reasoning and versatile mathematical skills. Practical exercises like these encourage understanding rather than rote learning, fostering deeper mathematics comprehension. They lay the groundwork for advanced topics, enriching students' overall mathematical literacy.