Problem 42 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (-\sqrt{2}, \sqrt{2}) $$

Step-by-Step Solution

Answer

The polar coordinates are \((2, \frac{3\pi}{4})\).

Step 1: Identify Rectangular Coordinates

The given rectangular coordinates are \[ (-\sqrt{2}, \sqrt{2}) \]which represent a point in the Cartesian coordinate system.

Step 2: Calculate r (the radial coordinate)

The radial coordinate \(r\) can be found using the formula for the distance from the origin: \[ r = \sqrt{x^2 + y^2} \]Substitute \(x = -\sqrt{2}\) and \(y = \sqrt{2}\) into the formula: \[ r = \sqrt{(-\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \]

Step 3: Calculate \(\theta\) (the angular coordinate)

To find the angle \(\theta\), use the tangent function: \[ \tan \theta = \frac{y}{x} \]Substitute \(x = -\sqrt{2}\) and \(y = \sqrt{2}\) into the formula: \[ \tan \theta = \frac{\sqrt{2}}{-\sqrt{2}} = -1 \]The reference angle where tangent is -1 is \(-\frac{\pi}{4}\). Since the point is in the second quadrant, \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

Step 4: Express the point in polar coordinates

The polar coordinates are expressed as \((r, \theta)\).From the previous steps, we have \(r = 2\) and \(\theta = \frac{3\pi}{4}\).The polar coordinates for the given rectangular coordinates are:\[ (2, \frac{3\pi}{4}) \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, allow us to pinpoint a location on a plane using two values: the x-coordinate (horizontal position) and the y-coordinate (vertical position). When you look at a grid, x-values shift you left or right while y-values move you up or down. This method was named after René Descartes, who famously used it to marry geometry and algebra in a revolutionary way. In our given example, we have the rectangular coordinates

• x = -\(\sqrt{2}\)
• y = \(\sqrt{2}\)

which indicates a specific point in the Cartesian plane. Here, it's crucial to identify that the x-value is negative, placing the point in the second quadrant (left of the y-axis). Understanding these coordinates is key as it lays the groundwork for any conversion to other systems like polar coordinates.

Radial Coordinate

The radial coordinate, often denoted as \(r\), gives the distance from the origin to the point in the polar coordinate system. To find this distance for any point given as rectangular coordinates, we employ the Pythagorean theorem in its distance formula form. This formula calculates \(r\) as:

• \(r = \sqrt{x^2 + y^2}\)

In our scenario with coordinates \((-\sqrt{2}, \sqrt{2})\), substituting the values results in:

• \(r = \sqrt{(-\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2\)

This calculation tells us the length of the straight line from the origin (0,0) to our point, irrespective of its location on the grid. A larger \(r\) value indicates a point further from the origin.

Angular Coordinate

The angular coordinate, represented as \(\theta\), specifies the angle between the positive x-axis and the line connecting the origin to the point. It defines the direction of the point from the origin in a counter-clockwise rotation. To determine \(\theta\), use the tangent function, since \(\tan\theta = \frac{y}{x}\).

• With \((-\sqrt{2}, \sqrt{2})\), \(\tan\theta = \frac{\sqrt{2}}{-\sqrt{2}} = -1\).

This corresponds to an angle where the tangent is -1. Here, we're interested in the reference angle associated with \(-\frac{\pi}{4}\), but because we're situated in the second quadrant, where tangent values are negative, we correct \(\theta\) to:

• \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\)

The final angular coordinate allows for precise direction placement in the polar map, completing the orientation of the point.

Cartesian Coordinate System

The Cartesian coordinate system is fundamental in mathematics for plotting points on a two-dimensional plane. Through intersecting horizontal (x-axis) and vertical (y-axis) lines, it forms a framework for locating points via pairs of numbers, called Cartesian or rectangular coordinates.

• The x-axis runs horizontally, while the y-axis is vertical
• The zero point, or origin, is at the intersect of both axes (0,0)

Introduced by René Descartes, this system revolutionized mathematical expressions by enabling algebraic equations to be interpreted visually as geometric shapes. For example, plotting \((-\sqrt{2}, \sqrt{2})\) on this grid positions our point in relation to these axes, facilitating conversions like those to polar coordinates. Essentially, recognizing these coordinates allows for multi-dimensional concepts to be shown clearly on a 2D surface.