Problem 27 - Calculus Early Transcendentals

Question

\(27-28\) For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a) A line through the origin that makes an angle of \(\pi / 6\) with the positive \(x\) -axis (b) A vertical line through the point \((3,3)\)

Step-by-Step Solution

Answer

(a) Polar equation: \( \theta = \frac{\pi}{6} \); Cartesian: \( y = \frac{1}{\sqrt{3}}x \). (b) Cartesian equation: \( x = 3 \).

Step 1: Analyzing the Line Through the Origin

The first part of the exercise asks for a line through the origin that makes an angle of \( \frac{\pi}{6} \) with the positive \( x \)-axis. In polar coordinates, such a line can be easily described using the angle \( \theta \). A line through the origin is given by \( \theta = \text{angle with } x\text{-axis} \). Hence, \( \theta = \frac{\pi}{6} \) is the polar equation for this line.

Step 2: Writing the Equation for the Line through the Origin

To express this line in Cartesian coordinates, remember that the slope, \( m \), of a line making an angle \( \alpha \) with the positive \( x \)-axis is \( \tan(\alpha) \). Thus, the slope is \( \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} \). The equation of the line through the origin in slope-intercept form is \( y = mx \), which becomes \( y = \frac{1}{\sqrt{3}}x \).

Step 3: Analyzing the Vertical Line through \((3,3)\)

In the second part of the exercise, we are dealing with a vertical line through the point \((3,3)\). In Cartesian coordinates, vertical lines are easily described. The equation of a vertical line is \( x = \text{constant} \), where the constant is the \( x \)-coordinate of the point it passes through. Hence, the Cartesian equation is \( x = 3 \).

Step 4: Polar Representation for Vertical Lines

Vertical lines are more easily described in Cartesian coordinates as they are straight lines parallel to the \( y \)-axis. Polar equations are generally less straightforward for vertical lines, as their expression would be more complex than the simple Cartesian form \( x = 3 \).

Key Concepts

Polar Coordinates

Polar coordinates are a method of representing points in a 2D plane via a radius and an angle. Unlike Cartesian coordinates which use a grid system, polar coordinates define a point based on its distance from the origin, known as the radius ( ) and the angle ( heta) measured from the positive x-axis. This form of representation is useful for curves that are naturally circular or spiral. It's a good choice for describing lines through the origin that have a specific angle with the x-axis.

• In polar coordinates, an angle \(\theta\) can describe a line through the origin. For instance, a line passing through the origin at an angle \(\frac{\pi}{6}\) is simply \(\theta = \frac{\pi}{6}\).
• Polar equations are particularly advantageous for representing curves like circles, spirals, and roses, where specifying a radius and angle is simpler than using Cartesian equations.

Understanding how to switch between polar and Cartesian coordinates gives greater flexibility in solving mathematical problems.

Cartesian Coordinates

Cartesian coordinates use a grid structure to define a point's position based on x and y values. This is a straightforward system that's typically used for tasks that involve rectangular or grid-like spatial understanding. Cartesian coordinates are well-suited to describe vertical, horizontal, and linear paths without an origin anchor.

• In situations like a vertical line through a specific point, such as (3,3), the Cartesian format \(x = 3\) succinctly describes the line's path.
• For linear relationships, such as expressing the relationship between two variables, a Cartesian approach is ideal.

Switching between coordinate systems can be necessary, but Cartesian coordinates offer clear advantages for certain problems, like vertical lines or linear equations through specified points.

Linear Equations

Linear equations describe a straight-line graph. They can be expressed in different forms such as slope-intercept form \(y = mx + b\) and point-slope form \((y - y_1) = m(x - x_1)\). These forms help understand how the line behaves. Cartesian equations often describe these effectively.

• A line through the origin with a slope derived from the angle is a typical linear problem, defined mathematically by \(y = mx\) where \(m\) is the slope.
• For example, a line passing through the origin with an angle \(\frac{\pi}{6}\) with the positive x-axis translates into a simple linear equation \(y = \frac{1}{\sqrt{3}}x\).
• Conversely, if a line runs vertically, the equation simplifies to \(x = c\) in Cartesian coordinates, where \(c\) is constant, such as in a line through \((3,3)\).

Mastering linear equations involves understanding the different forms and situating within coordinate systems to denote lines accurately.