Problem 41 - Precalculus Mathematics for Calculus

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(-6,0)$$

Step-by-Step Solution

Answer

The polar coordinates are \((6, \pi)\).

Step 1: Determine the Value of r

The formula to convert from rectangular coordinates \((x,y)\) to polar coordinates \((r, \theta)\) is: \[r = \sqrt{x^2 + y^2}\] Substitute \(x = -6\) and \(y = 0\) into the formula: \[r = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6\]So, \(r = 6\).

Step 2: Determine the Value of θ

The angle\(\theta\) in polar coordinates can be found using the arctangent function. Since \(y = 0\),\(\theta\) is along the x-axis. Since the point is \((-6, 0)\) and all on the negative x-axis, \(\theta\) is \( \pi \).So, \(\theta = \pi\).

Step 3: Write the Polar Coordinates

Combine the values of \(r\) and \(\theta\) to write the polar coordinates. The polar coordinates are \((r, \theta) = (6, \pi)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way of representing points on a plane using two numbers, usually written as \((x, y)\). These numbers denote the horizontal and vertical distances from a reference point, commonly known as the origin. The origin is where the horizontal axis (x-axis) and the vertical axis (y-axis) intersect, marked as the point \((0, 0)\).

• The x-coordinate indicates how far left or right the point is from the origin. Positive x-values are to the right, and negative x-values are to the left.
• The y-coordinate shows how far up or down the point is in relation to the origin. Positive y-values are above, and negative y-values are below.

Rectangular coordinates are useful because they allow us to easily plot points and visualize their locations on a graph. They serve as the foundation for geometric calculations and transformations.

Conversion Formulas

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use specific conversion formulas. The idea is to transform the way we describe the location of a point, from Cartesian to a circular format.

• The formula for the radial distance \(r\) is: \[r = \sqrt{x^2 + y^2}\]This represents the straight line distance from the origin to the point.
• The formula for the angle \(\theta\) is: \[\theta = \tan^{-1} \left( \frac{y}{x} \right)\]This gives the direction angle of \(r\) from the positive x-axis.

Keep in mind to adjust \(\theta\) appropriately depending on the quadrant in which the point resides.

These formulas help in navigating between different coordinate systems, which can be very useful in applications like physics or engineering.

Angle Calculation

Calculating the angle \(\theta\) in polar coordinates often involves using trigonometric functions. With the tangent function \(\tan(\theta)\), we can determine the tangent of the angle formed by the point with respect to the positive x-axis.

In our example, \((-6, 0)\) lies on the x-axis, and most of the trigonometric calculations simplify significantly:

• When the y-coordinate is zero, the angle is aligned directly with the x-axis. This is a special case where the calculation is straightforward.
• For \(x < 0\) (as in our point), the angle becomes \(\theta = \pi\), which points exactly along the negative x-axis.

Understanding angle calculations is crucial, making it easier to visualize the directional relationship between the point and the origin.

Coordinate System

Coordinates are part of a larger coordinate system, which provides a framework for locating points on a plane. The most common systems are the rectangular (or Cartesian) and polar coordinate systems.

• In the rectangular system, points are described using perpendicular axes, as we've explored with \((x, y)\) coordinates.
• In the polar system, points are represented by their radial distance from the origin \(r\) and their angle from the positive x-axis \(\theta\).

Using different systems can simplify many problems. For example, circular motion is easier to describe with polar coordinates. Coordinate systems allow us to tailor the description of a point depending on the problem at hand, providing flexibility and insight.