Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$(-\sqrt{3}, 2)$$

Step-by-Step Solution

Answer

The polar coordinates for the point (-√3, 2) are (\sqrt{7}, 2.094).

Step 1: Understanding the Generic Conversion Process

The first thing to understand is that we can convert rectangular to polar coordinates using these generic formulas: \( r = \sqrt{x^2 + y^2} \) and \( \theta = arctan(\frac{y}{x}) \). Where \( r \) is the distance from origin to the point, \( x \) and \( y \) are rectangular coordinates, and \( \theta \) is the angle made by the line joining point to the origin with respect to x-axis in anti-clockwise direction.

Step 2: Substituting given values into the Formulas

Given that \( x = -\sqrt{3} \) and \( y = 2 \), substituting these values into the formula for \( r \), we get \( r = \sqrt{(-\sqrt{3})^2 + 2^2} = \sqrt{3 + 4} = \sqrt{7} \). Now we substitute \( x = -\sqrt{3} \) and \( y = 2 \) into the formula for \( \theta \) : \( \theta = arctan(\frac{2}{-\sqrt{3}}) \). Using a graphing utility, or by looking at the trigonometric signs in various quadrants, one can tell that the angle lies in the second quadrant where tan function is positive.

Step 3: Converting Angle from Trigonometric Form

We may need to adjust the angle based on the sign of \( y \) and \( x \). Since in the second quadrant, both sine and cosine are negative, to obtain the appropriate \( \theta \) with reference to positive x-axis, we add \( \pi \) to the calculated \( \theta \). Therefore, if we calculate \( \theta = arctan(\frac{2}{-\sqrt{3}}) \), we get \( \theta \approx -1.047 \), and adding \( \pi \) to it gives: \( \theta = -1.047 + \pi \approx 2.094 \) radians.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent a point in a plane using two values, typically labeled as \( x \) and \( y \). These coordinates reflect the position of the point by its horizontal (\( x \)) and vertical (\( y \)) distances from a fixed origin.

For example, consider the point \((-\sqrt{3}, 2)\). Here, \(x = -\sqrt{3}\) and \(y = 2\). This tells us that the point is located \(\sqrt{3}\) units to the left of the origin (since the \( x \) value is negative) and 2 units above the origin (as the \( y \) value is positive).

Rectangular coordinates are commonly used in mathematics for describing positions in both geometry and algebra. They are easy to work with, especially when you need direct measures of distances parallel to the two primary axes.

Graphing Utility

A graphing utility is a tool that aids in visualizing mathematical functions and figures, such as graphs of equations or geometric shapes. It can be a physical calculator or a software application that allows users to input equations and instantly see the graphical representation.

When converting rectangular coordinates like \((-\sqrt{3}, 2)\) to polar coordinates, using a graphing utility can help visualize the position of the point in a polar graph.

• It assists in identifying the correct quadrant where the point lies.
• Helps calculate and verify angles such as \(\theta\) using arctan calculations.
• Allows for quick adjustments if errors are observed in manual calculations.

Graphing utilities are invaluable for students because they provide immediate graphical feedback, helping ensure the calculations are logical in spatial reasoning.

Trigonometric Functions

Trigonometric functions, like sine, cosine, and tangent, play a crucial role in converting rectangular coordinates to polar coordinates. To understand this conversion, recall that polar coordinates are described by \(r\) (the radial distance from the origin) and \(\theta\) (the angle from the positive x-axis).

For instance, using the equation \( r = \sqrt{x^2 + y^2} \), we calculate the radial distance from the origin for our point \((-\sqrt{3}, 2)\) as \( \sqrt{7} \). The angle \( \theta \) is determined by \( \tan^{-1}(\frac{y}{x}) \), where polar coordinates provide non-linear relationships in terms of rotation and magnitude of points.

• To adjust for specific quadrants, additional steps might be needed, such as adding \(\pi\) to \(\theta\) when \(x\) is negative.
• In our specific example, recognizing that the angle should reside in the second quadrant due to negative \(x\) and positive \(y\) is crucial.
• Graphing utilities can cross-check these angles on unit circles in real-time.

Mastering these functions empowers students to interpret angles and distances with precision, useful in broad applications from physics to engineering.