Problem 51 - Algebra and Trigonometry

Question

Convert each rectangular equation to a polar equation that expresses r in terms of \(\theta\). $$ x=7 $$

Step-by-Step Solution

Answer

The polar equation expressing \(r\) in terms of \(\theta\) from the rectangular equation \(x=7\) is \(r=\frac{7}{\cos(\theta)}\).

Step 1: Write down the given equation

The given rectangular equation is \(x = 7\).

Step 2: Write down the polar coordinate system relation

In polar coordinates, the relation between \(x\) and \(r\) is given by \(x = r\cos(\theta)\).

Step 3: Substitute the value of \(x\) from the given rectangular equation into the polar coordinate relation

Substitute \(x = 7\) into \(x = r\cos(\theta)\) to get \(7 = r\cos(\theta)\).

Step 4: Solve for \(r\)

After substitution, the equation we have is \(7 = r\cos(\theta)\). In order to express \(r\) in terms of \(\theta\), we need to divide both sides of the equation by \(\cos(\theta)\). This gives the polar equation \(r = \frac{7}{\cos(\theta)}\).

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system where points are described using two perpendicular axes called the x-axis and the y-axis.
In this system, any point in a plane can be represented by an ordered pair \( (x, y) \). The first number in the pair measures how far a point is along the x-axis, and the second number measures how far it is along the y-axis.

• The x-axis runs horizontally and determines the point's horizontal distance from the origin.
• The y-axis runs vertically and measures the point's vertical distance from the origin.

For example, the point \( (7, 0) \) means that the point is 7 units away from the y-axis on the positive side of the x-axis, and it is positioned exactly on the x-axis since the y-coordinate is 0.
Rectangular coordinates are widely used in everyday mathematics because of their simplicity and direct relationship with addition and subtraction along the two axes.

An Introduction to Trigonometric Functions

Trigonometric functions are mathematical tools that relate the angles of a triangle to the lengths of its sides. These functions are fundamental in the study of periodic phenomena such as sound and light waves.
There are six main trigonometric functions, but for the conversion from rectangular to polar coordinates, cosine and sine are most relevant.

• Cosine \( \cos \theta \) takes an angle \( \theta \) and returns the ratio of the adjacent side to the hypotenuse in a right triangle.
• Sine \( \sin \theta \) gives the ratio of the opposite side to the hypotenuse.

In the context of coordinates, these functions help in expressing how x and y are related to a point’s distance from the origin (r) and its angle \( \theta \). For example, if the polar equation for r is \( r \cos \theta \) = x, then x is the horizontal projection of r on the x-axis.
Trigonometric functions enable us to convert between different coordinate systems seamlessly.

Coordinate Conversion Between Rectangular and Polar Systems

Coordinate conversion is the process of changing a point's representation from the rectangular (Cartesian) system to the polar coordinate system and vice versa. This is essential when solving mathematical problems in different contexts such as physics, engineering, or navigation.

• In the polar coordinate system, a point is represented by \( (r, \theta) \), where r is the distance from the origin and \( \theta \) is the angle it makes with the positive x-axis.
• To convert from rectangular coordinates to polar, we use the equations: \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1} (\frac{y}{x}) \).
• To convert from polar to rectangular, we use \( x = r \cos \theta \) and \( y = r \sin \theta \).

In our original example \( x = 7 \), we used the conversion formula \( x = r \cos \theta \) to derive the polar equation \( r = \frac{7}{\cos \theta} \).
This conversion highlights how different systems provide various insights into mathematical relationships and geometries, often simplifying complex problems or providing new perspectives.