Problem 56 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=\frac{4}{1+2 \sin \theta} $$

Step-by-Step Solution

Answer

The rectangular equation is \(x^2 - 3y^2 + 16y - 16 = 0\).

Step 1: Understand Polar to Rectangular Conversion

In polar coordinates, a point is represented as \((r, \theta)\). The corresponding rectangular coordinates \((x, y)\) can be found using the relationships: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r = \sqrt{x^2 + y^2}\) and \(\tan \theta = \frac{y}{x}\).

Step 2: Identify Known Values

We are given the polar equation: \(r = \frac{4}{1 + 2\sin\theta}\). We need to convert this equation to rectangular form. To do this, identify where in the equation we can substitute known substitutes from the conversion relationships.

Step 3: Substitute for \(r\) in the Equation

Since \(y = r\sin\theta\), we also know that \(\sin\theta = \frac{y}{r}\). Substitute \(\sin\theta\) in the given equation: \[r = \frac{4}{1 + 2\left(\frac{y}{r}\right)} = \frac{4}{1 + \frac{2y}{r}}\]. Multiply both sides by \(1 + \frac{2y}{r}\): \[r \left(1 + \frac{2y}{r}\right) = 4\]. Simplify: \[r + 2y = 4\].

Step 4: Solve for \(r\)

Reorganize the equation: \[r = 4 - 2y\]. Since \(r\) represents the radial distance and can be expressed as \(\sqrt{x^2 + y^2}\), substitute: \[\sqrt{x^2 + y^2} = 4 - 2y\].

Step 5: Square Both Sides

To eliminate the square root, square both sides of the equation: \[(\sqrt{x^2 + y^2})^2 = (4 - 2y)^2\]. This results in: \[x^2 + y^2 = (4 - 2y)^2\].

Step 6: Expand the Right Side

Expand \((4 - 2y)^2\): \[4^2 - 2 \cdot 4 \cdot 2y + (2y)^2 = 16 - 16y + 4y^2\]. This gives: \[x^2 + y^2 = 16 - 16y + 4y^2\].

Step 7: Rearrange into Standard Form

Rearrange to group like terms: \[x^2 + y^2 - 4y^2 + 16y - 16 = 0\]. Simplify: \[x^2 - 3y^2 + 16y - 16 = 0\]. This is the equation in rectangular coordinates.

Key Concepts

polar coordinates

Polar coordinates are a way to represent points on a plane using two values: the radial distance from a fixed point (origin) and an angle from a fixed direction. The radial distance is denoted as \( r \) and the angle as \( \theta \). This system is particularly helpful when dealing with problems involving circular or rotational symmetry.

Imagine a point in space. Instead of describing its position using a grid with the usual x and y axes, you can describe how far away it is from a starting point (r) and how much you need to rotate an arm from a reference direction (theta) to reach it.

• \( r \) - Radial distance from the origin.
• \( \theta \) - Angle from a reference direction, usually the positive x-axis.

Polar coordinates can be advantageous in scenarios involving circular motion, waves, and vibrations, making them very useful in physics and engineering applications.

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a well-known method for specifying a point's location on a plane using two numerical values: \( x \) and \( y \). These values correspond to horizontal and vertical positions relative to two perpendicular number lines (axes).

With rectangular coordinates, you can picture the plane as a giant sheet of graph paper. Each point on this sheet is defined by how far it is from the x-axis and the y-axis.

• \( x \) - Distance along the horizontal axis.
• \( y \) - Distance along the vertical axis.

Using rectangular coordinates is convenient for many mathematical applications, especially when dealing with linear systems, calculating distances, and performing integrals over rectangular areas.

coordinate transformation

Coordinate transformation is the process of converting one type of coordinate system into another. In the context of this particular exercise, we are focusing on converting polar coordinates to rectangular ones.

This transformation is essential for interpreting problems in contexts where different coordinate systems are more intuitive or beneficial. The polar to rectangular transformation relies on trigonometric relationships:

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

These equations enable the conversion between the systems, allowing the representation of equations and functions in a more useful form depending on the task.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They play a key role in mathematics, especially during the process of coordinate conversion.

When converting from polar to rectangular coordinates, certain trigonometric identities can simplify the transformation process and help solve equations. The identities frequently used are:

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

These identities work as shortcuts and checks when manipulating trigonometric expressions, enabling a smoother transition between different forms and ensuring accuracy in calculations. Understanding and effectively using these identities are crucial, especially when dealing with more complex trigonometric transformations.