Problem 10 - Thomas Calculus

Question

Find a polar equation in the form \(r \cos \left(\theta-\theta_{0}\right)=r_{0}\) for each of the lines in Exercises \(9-12\) . $$ \sqrt{3} x-y=1 $$

Step-by-Step Solution

Answer

The polar equation is: \( r \cos(\theta - \frac{\pi}{3}) = 1 \).

Step 1: Identify the Cartesian equation form

The given line is in the Cartesian form \(\sqrt{3}x - y = 1\).Recognize that this represents a linear equation.

Step 2: Convert to slope-intercept form

The slope-intercept form of a line equation is \(y = mx + c\). Therefore, rearrange the given equation:\(y = \sqrt{3}x - 1\). Here, \(m = \sqrt{3}\) represents the slope of the line, and \(c = -1\) represents the y-intercept.

Step 3: Identify angle with the x-axis

The slope \(m = \sqrt{3}\) corresponds to the tangent of the angle the line makes with the positive x-axis. Use the tangent function:\(\tan(\theta) = \sqrt{3}\).This angle \(\theta\) is \(= \frac{\pi}{3}\), because \(tan^{-1}(\sqrt{3}) = \frac{\pi}{3}\).

Step 4: Express the line in polar form

Using the polar equation for a line, \(r \cos \left(\theta - \theta_0\right) = r_0\), where \(\theta_0\) is the angle with the x-axis:Given \(\theta_0 = \frac{\pi}{3}\), the line equation is:\(r \cos(\theta - \frac{\pi}{3}) = 1\). Here, \(r_0\) corresponds to the constant term in the original equation.

Key Concepts

Cartesian to Polar Conversion

When dealing with mathematical equations, it's important to understand both Cartesian and polar coordinates. Cartesian coordinates are given by \(x\) and \(y\), representing positions in a rectangular grid. However, polar coordinates, represented by \(r\) and \(\theta\), focus on the distance and angle from a reference point, often the origin.

Conversion from Cartesian to polar involves transforming the \((x, y)\) coordinates of a point into \(r\) and \(\theta\). The formulae for this conversion are:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1} \left( \frac{y}{x} \right)\)

This practical approach helps in solving problems related to circular and spiraling movements, commonly used in physics and engineering. Knowing how to convert between these systems is critical for understanding complex geometrical and mathematical scenarios.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It facilitates identifying the line's slope and y-intercept directly from the equation. This form is given by: \(y = mx + c\)

In this equation:

• \(m\) represents the slope, or steepness, of the line. It tells us how much \(y\) increases or decreases as \(x\) increases by one unit.
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

For example, in the equation \(y = \sqrt{3}x - 1\), \(m\) = \sqrt{3}\ and \(c\) = -1. This means the line goes up \sqrt{3}\ units for every 1 unit it goes right and cuts the y-axis at -1. This form is incredibly useful for graphing lines and calculating the exact values of intersection points.

Tangent Function

The tangent function is a key concept in trigonometry, used to describe the slope of an angle within a right triangle. Given a right triangle, the tangent of an angle \(\theta\) is observed by:

• \(\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\)

The tangent function relates directly to the slope of a line in trigonometry. In a linear equation, the coefficient representing the slope \(m\) often equals \( an(\theta)\). For instance, if \(m = \sqrt{3}\), then \(\tan(\theta) = \sqrt{3}\), which leads to \(\theta = \frac{\pi}{3}\) as it corresponds to a standard angle in trigonometry.

The tangent function is essential for identifying angles in various applications, making it crucial for understanding slopes and angles in geometry and applied mathematics.

Angle Identification

Identifying the angle a line makes with the x-axis is an important aspect of understanding the geometry of lines. This involves knowing the slope, or tangent, of the angle. The angle \(\theta\) between a line and the x-axis can be determined using the arctangent function, \(\tan^{-1}\).

For a line with slope \(m = \tan(\theta)\), \( heta = \tan^{-1}(m)\). For example, if \(m = \sqrt{3}\), then \( heta = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}\). This angle \(\theta\) helps in expressing a line in polar form, where the angle \(\theta_0\) is utilized.

• Understanding angles is crucial in trigonometric calculations.
• It provides a visual and calculative representation of direction.

Accurately identifying angles is vital for constructing geometric solutions and applying these in real-world contexts.

Linear Equations

Linear equations are algebraic expressions that describe a straight line on a Cartesian plane. They form the basis of much of algebra and geometry. These equations typically appear in the form \(Ax + By = C\), where \(A, B,\) and \(C\) are constants.

Such equations can easily be rearranged into the slope-intercept form \(y = mx + c\), making them useful for graphing and analytical purposes. Linear equations have a constant rate of change, which means they model scenarios where the relationship between variables is proportional and consistent.

• They are foundational in understanding the relationships between variables in mathematics.
• They provide straightforward calculations for predictions and projections in various disciplines.

Understanding linear equations is essential in math, as they are widely applicable in fields like physics, economics, and engineering.