Problem 13 - Precalculus

Question

Convert to rectangular form. $$r^{2}=8 /\left(2-\sin ^{2} \theta\right)$$

Step-by-Step Solution

Answer

The rectangular form of the equation is \(2x^2 + y^2 = 8\).

Step 1: Recall Polar to Rectangular Conversion

To convert from polar to rectangular coordinates, use the following relationships:- \( r = \sqrt{x^2 + y^2} \)- \( x = r \cos(\theta) \)- \( y = r \sin(\theta) \)- \( \sin(\theta) = \frac{y}{r} \)and \( \cos(\theta) = \frac{x}{r} \). These will come in handy later in the process.

Step 2: Rearrange Given Equation

We start with: \( r^2 = \frac{8}{2 - \sin^2 \theta} \).First, multiply both sides by \(2 - \sin^2 \theta\):\[ r^2 (2 - \sin^2 \theta) = 8 \]

Step 3: Simplify the Equation

Expand the left side: \[ 2r^2 - r^2 \sin^2 \theta = 8 \] Now, use the relation \( \sin(\theta) = \frac{y}{r} \) to express \( \sin^2(\theta) \) as \( \left(\frac{y}{r}\right)^2 \), which is \( \frac{y^2}{r^2} \).

Step 4: Substitute Polar Relationships

Substitute \( \sin^2(\theta) \) with \( \frac{y^2}{r^2} \) in the equation:\[ 2r^2 - r^2 \left(\frac{y^2}{r^2}\right) = 8 \] which simplifies to:\[ 2r^2 - y^2 = 8 \]

Step 5: Substitute \( r^2 = x^2 + y^2 \)

Since \( r^2 = x^2 + y^2 \), substitute this into the equation:\[ 2(x^2 + y^2) - y^2 = 8 \] which simplifies to:\[ 2x^2 + y^2 = 8 \]

Step 6: Final Rectangular Form Equation

The final equation in rectangular coordinates is:\[ 2x^2 + y^2 = 8 \] This equation represents a conic section (an ellipse) in rectangular form.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system in mathematics where a pair of values determines the position of a point in space. These coordinates consist of two numbers: the x-coordinate (horizontal position) and the y-coordinate (vertical position). Together, they form a point \(x, y\) on a two-dimensional plane.

Understanding the conversion from polar coordinates to rectangular coordinates involves using the conversion formulas:

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

These relationships allow for translating the circular, radial description of a point in terms of angle (\(\theta\)) and radius (\(r\)) into a linear, grid-based system of measurement using x and y axes.

Exploring conversions such as these helps in visualizing mathematical concepts beyond simple arithmetic calculations, paving the way for geometrically interpreting complex problems using algebra.

Conic Sections

Conic sections are curves obtained by slicing a right circular cone with a plane. They are essential shapes in mathematics and appear frequently in algebra and geometry. The principal types of conic sections include ellipses, circles, hyperbolas, and parabolas.

In this exercise, the rectangular coordinate equation derived is \( 2x^2 + y^2 = 8 \), which represents an ellipse. To distinguish between different conics, it's important to analyze the general form of a conic section equation: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), where various factors determine the shape:

• Ellipse: \( A > 0, C > 0 \) and \( AC > B^2 \)
• Parabola: \( B^2 = AC \)
• Hyperbola: \( AC < B^2 \)

The coefficients influence the orientation and dimensions of the conic section. Recognizing these forms aids in understanding the geometry and properties of the conics, and how they fit into the broader context of mathematical analysis.

Polar Coordinates

Polar coordinates provide a method for describing points in terms of their distance from a central point (radius \(r\)) and the angle \(\theta\) from the positive x-axis. Unlike rectangular coordinates which use a grid system, polar coordinates employ a radial approach, making them ideal for dealing with circular or rotational symmetry.

In polar coordinate conversions, it's essential to grasp the link between \(r\) and \(\theta\) and how they map to rectangular coordinates through formulas like \( r = \sqrt{x^2 + y^2} \), among others. This allows the translation from polar to rectangular forms and vice versa.

Such conversions are indispensable in applications involving curves and paths that naturally fit a radial symmetry, like conic sections. Naturally, using polar coordinates simplifies computations in scenarios where symmetry around a point is prominent, aiding in the analysis and understanding of spatial phenomena.