Problem 41 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ y=7 x $$

Step-by-Step Solution

Answer

The polar equation is \( \theta = \arctan(7) \).

Step 1: Understanding the Rectangular to Polar Conversion

To find a polar equation that is equivalent to the given rectangular equation, we need to express the equation in terms of polar coordinates. In the polar coordinate system, the relationships between polar coordinates \(r, \theta\) and rectangular coordinates \(x, y\) are given by: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Substitute Variables

Substitute \(y = r \sin \theta\) and \(x = r \cos \theta\) into the given rectangular equation \(y = 7x\):\[r \sin \theta = 7(r \cos \theta)\]

Step 3: Simplify the Equation

Now, simplify the equation obtained in Step 2:\[r \sin \theta = 7 r \cos \theta\]Divide both sides by \(r\) (assuming \(r eq 0\)):\[\sin \theta = 7 \cos \theta\]

Step 4: Express in Standard Polar Form

Transform \(\sin \theta = 7 \cos \theta\) into a standard form for polar equations by dividing both sides by \(\cos \theta\):\[\tan \theta = 7\]Since this represents a line in polar coordinates where \(\theta = \arctan(7)\), the equivalent polar equation reflects the line's slope.

Key Concepts

Polar Coordinates

In mathematics, polar coordinates are a way of representing points on a plane. Instead of using the traditional rectangular (or Cartesian) coordinates, which employ an x and a y-axis, polar coordinates locate a point based on its distance from a fixed reference point, called the origin, and the angle it forms with a reference direction, usually the positive x-axis.

**Key Components of Polar Coordinates**:

• Radius (r): The distance from the origin to the point. It tells how far the point is located from the center of the system.
• Angle (θ): This is the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin with the point. The unit of this angle is typically radians.

The polar coordinate system is extremely useful in scenarios involving circles or spirals, and simplifies complex computations in trigonometry and calculus by reducing multi-dimensional problems to one-dimensional functions.

Rectangular to Polar Conversion

Converting rectangular coordinates to polar coordinates is a common task in mathematics, especially when dealing with equations involving circles or other curves. This conversion allows the expression of equations in the polar coordinate system, often making it easier to identify certain properties of the graphs.

**Conversion Formulas**:

• From Rectangular to Polar: Given rectangular coordinates \(x, y\), the conversion formulae are:
• \( r = \sqrt{x^2 + y^2} \) - This formula calculates the distance from the origin, combining both the x and y values.
• \( \theta = \arctan(\frac{y}{x}) \) - This angle is determined by the inverse tangent function and represents the direction of the point from the x-axis.

In the specific problem given \(y = 7x\), we substitute \(y = r \sin \theta\) and \(x = r \cos \theta\) into the equation to transform it into the polar equation \( \tan \theta = 7 \), highlighting its line in polar coordinates where the angle is constant.

Trigonometric Functions

A crucial part of working with polar coordinates and converting equations is the use of trigonometric functions. These functions allow the translation of coordinates from one system to the other, through the inherent relationships between angles and lengths.

**Important Trigonometric Functions**:

• Sine (sin) and Cosine (cos): These functions relate angles to the x and y coordinates, where \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \). These relationships help in identifying the y and x placements within the unit circle.
• Tangent (tan): Expressed as \( \tan \theta = \frac{y}{x} \), it is vital in polar conversions, particularly when reducing \( \sin \theta = \frac{y}{x} \cos \theta \) to \( \tan \theta \).

In our exercise, simplifying the equation \( r \sin \theta = 7 r \cos \theta \) directly leads us to the tangent relation \( \tan \theta = 7 \), where the use of trigonometric functions helps establish the line's direction and angle in its polar form.