Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the Cartesian equations of the graphs of the given polar equations. $$ \theta=\frac{1}{2} \pi $$

Step-by-Step Solution

Answer

The Cartesian equation is \( x = 0 \).

Step 1: Understanding Polar Equation

The given polar equation is \( \theta = \frac{1}{2} \pi \). This equation means that all the points on this graph have an angle of \( \frac{1}{2} \pi \) or 90 degrees from the positive x-axis.

Step 2: Identify the Nature of the Graph

Since \( \theta = \frac{1}{2} \pi \) is a fixed angle, the graph represents a line that is perpendicular to the initial line (the positive x-axis) and goes through the origin. This is a vertical line in the Cartesian plane.

Step 3: Convert to Cartesian Coordinates

In polar coordinates, if \( \theta \) is fixed and equals \( \frac{1}{2} \pi \), the line is vertical and along the y-axis. In Cartesian coordinates, this line is equivalent to the line \( x = 0 \).

Step 4: Concluding the Cartesian Equation

Given that the polar graph is a vertical line, its Cartesian equation is simply \( x = 0 \), representing all points where \( x \) is zero, which means it lies on the y-axis.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to locate points in a plane using two distinct values: the radial distance and the angle from a reference direction. Instead of using the traditional Cartesian "x" and "y" coordinates, polar coordinates use "r" (the radius) and "θ" (the angle).

• The radius "r" represents the distance from the origin (or center) to the point.
• The angle "θ" measures the direction of the point from the positive x-axis.

These coordinates are often useful in situations involving circular or rotational symmetry, because they can make the mathematics simpler compared to Cartesian coordinates. For example, a circle centered at the origin is just described by its radius "r", without the need for complex equations. When dealing with angles like \( \theta = \frac{1}{2} \pi \), it signifies a 90-degree angle from the positive x-axis, leading to simpler interpretations such as vertical or horizontal lines.

Cartesian Coordinates

In a Cartesian coordinate system, each point is defined by two numbers, "x" and "y", which denote its position on a 2-dimensional grid. This system is widely used in algebra and calculus because it provides a straightforward way to represent equations and geometric shapes.

• The "x" coordinate specifies the horizontal position of a point.
• The "y" coordinate specifies the vertical position.

The grid is made up of perpendicular lines and is represented by two axes: the x-axis (horizontal) and the y-axis (vertical). Cartesian coordinates simplify translating complex geometrical shapes and functions into mathematical equations. For instance, the equation \( x = 0 \) implies a vertical line through the y-axis, highlighting the versatility of this system.

Vertical Line Equation

A vertical line in the Cartesian coordinate system has a unique property: it features a constant x-value for all points on the line. This results in an equation of the form \( x = k \), where "k" is the constant x-coordinate.

• In the given example, the vertical line is described by \( \theta = \frac{1}{2} \pi \), which leads to the line \( x = 0 \) in Cartesian coordinates.
• Vertical lines run parallel to the y-axis and do not have any slope because they do not rise in the "y" direction relative to the "x" direction.

Understanding vertical lines is important because they are fundamental to topics such as linear equations and graphing. Their simplicity, with no need to account for the y-coordinate beyond its freedom to vary, highlights their significance in various mathematical contexts.

Angle Conversion

Converting angles from polar to Cartesian contexts involves understanding the interplay between radians and degrees. In polar coordinates, angles are often expressed in radians, a measure based on the radius of a circle.

• The conversion between degrees and radians is essential for many mathematical applications.
• For instance, \( \theta = \frac{1}{2} \pi \) radians is equivalent to 90 degrees.

This 90-degree angle is crucial in various scenarios, such as determining the direction of lines or vectors. Mastering angle conversion can make it easier to transition between polar and Cartesian coordinates, enhancing your understanding of trigonometric functions and geometric interpretations.