Problem 24 - Calculus Early Transcendentals

Question

Express the following Cartesian coordinates in polar coordinates in at least two different ways. (-9,0)

Step-by-Step Solution

Answer

The given Cartesian coordinates (-9,0) can be represented in polar coordinates as (9, π) and (9, 3π).

Step 1: Convert to polar coordinates (method 1)

To convert (-9,0) to polar coordinates, we first calculate 'r'. Since x = -9 and y = 0, we have r = √((-9)²+0²) = 9. Next, we find the angle θ. Since y = 0, the point lies on the x-axis. The angle θ is therefore 180°, or π in radians. So, the polar coordinates for (-9,0) are (9, π).

Step 2: Convert to polar coordinates (method 2)

We can also represent the same point using another value for θ. If we increase the angle by 360° (2π radians), we would still be at the same point. So, another set of polar coordinates for (-9,0) would be: (9, π + 2π) = (9, 3π). Thus, the two different ways to express the Cartesian coordinates (-9,0) in polar coordinates are (9, π) and (9, 3π).

Key Concepts

Cartesian Coordinates

In the world of geometry, Cartesian coordinates are a way to specify a point in a two-dimensional space using a pair of numerical values. These values are usually labeled as \( x \) and \( y \). In a Cartesian plane, the horizontal axis is often called the \( x \)-axis, while the vertical axis is referred to as the \( y \)-axis.

For example, a point like \((-9, 0)\) in Cartesian coordinates means that we move 9 units left from the origin along the \( x \)-axis while staying at the origin along the \( y \)-axis. Cartesian coordinates are straightforward if you think about moving horizontally and vertically from a central point called the origin, where the \( x \)-axis and \( y \)-axis meet at (0, 0).

This system is incredibly useful for plotting points in algebra and is a fundamental concept in mathematics for describing the position of points in two dimensions.

Coordinate Conversion

Switching from Cartesian to polar coordinates involves a transformation that makes use of geometry and trigonometry.

Here's the step-by-step process:

• First, you calculate the distance from the origin to the point, which is known as \( r \) (the radial distance or radius). For a point \((x, y)\), the radius \( r \) is calculated using the formula: \[ r = \sqrt{x^2 + y^2}. \]
• Next, you determine the angle \( \theta \) (theta), which represents the direction from the origin to the point. For points on the \( x \)-axis, you may consider the angle based on standard trigonometric angles. For a point like \((-9, 0)\), the angle \( \theta \) can be found as 180° because it lies on the negative \( x \)-axis.

This transformation reveals how far a point is from the origin and the direction it makes with the positive \( x \)-axis.

When converting, remember that there can be multiple ways to represent the same point - by adding integers of 360° (or \( 2\pi \) radians) to \( \theta \), you can achieve equivalent polar coordinates. This process helps in understanding points in a circular context rather than a grid.

Radians

Radians are a unit of angular measure in mathematics, much like degrees. However, radians provide a more natural measure in calculus and trigonometry because they relate directly to the properties of circles.

One full revolution around a circle is \( 2\pi \) radians, which is equivalent to 360 degrees in the degree system. So, \( \pi \) radians represents a half-circle (or 180°). This unit emerges naturally due to the relationship between a circle's radius and its circumference, which is \( 2\pi \cdot r \).

Using radians allows us to express angles in terms of \( \pi \), which simplifies computations in advanced mathematics. For example, in the problem where the Cartesian coordinates \((-9,0)\) are given, we convert to polar coordinates and express the angle as \( \pi \) radians (180°).

• Radians are preferred in calculus because they simplify derivative and integral calculations on trigonometric functions.
• Understanding radians is crucial, as it provides a more profound comprehension of angular movement in mathematical formulas.

Radians might seem challenging at first, but practicing conversions between degrees and radians can greatly enhance comprehension.