Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Plot the point whose rectangular coondinates are given. Then determine nwo pairs of polar coondinates for the point with \(0^{\circ} \leq \theta<360^{\circ} .\) Do not use a calculator. $$(-1,1)$$

Step-by-Step Solution

Answer

The polar coordinates are \((\sqrt{2}, 135^\circ)\) and \((-\sqrt{2}, 315^\circ)\).

Step 1: Identify Rectangular Coordinates

The given rectangular coordinates are \((-1, 1)\). This means the point is located at a horizontal position of \(-1\) and a vertical position of \(1\) on the Cartesian plane.

Step 2: Plot the Point

To plot the point \((-1, 1)\), move 1 unit left along the x-axis from the origin, and 1 unit up along the y-axis. The point is located in the second quadrant of the Cartesian plane.

Step 3: Convert Rectangular to Polar Coordinates

To convert the point \((-1,1)\) to polar coordinates, calculate \(r\) and \(\theta\). The formula for \(r\) is \(r = \sqrt{x^2 + y^2}\). Substituting \(x = -1\) and \(y = 1\), calculate: \[r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}\]

Step 4: Determine the Angle \(\theta\)

The angle \(\theta\) is found using the arctangent function, \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).Since the point is in the second quadrant where x is negative and y is positive, the reference angle is:\[\theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) = -45^\circ\]However, angles in the second quadrant are in the range of \(90^\circ\) to \(180^\circ\). Therefore, convert it by adding \(180^\circ\):\[\theta = 180^\circ - 45^\circ = 135^\circ\]

Step 5: Provide Two Sets of Polar Coordinates

Now that we have the polar coordinates, one set is \((r, \theta) = (\sqrt{2}, 135^\circ)\). Some other polar coordinates canonically equivalent would be \(r\) negative with \(\theta + 180\) degrees, as moving 180 degrees is equivalent to reversing direction. Calculate:\[\text{Another set: } (-r, \theta + 180^\circ) = (-\sqrt{2}, 315^\circ)\]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often termed as Cartesian coordinates, are a fundamental concept in mathematics and physics. These coordinates help pinpoint the location of a point within a plane, using two numerical values. Typically, these values are denoted as \(x, y\). The \(x\) value describes a point's position horizontally, while the \(y\) value specifies its vertical position. For example, the coordinates (-1, 1) indicate:

• Move one unit horizontally left from the origin, because the value -1 is negative.
• Move one unit vertically up from the origin, because the value 1 is positive.

Combining these steps places the point in the second quadrant of the Cartesian plane. Hence, the rectangular coordinates offer a straightforward method to locate any point on a 2D plane by accounting for both horizontal and vertical distances.

Cartesian Plane

The Cartesian plane, named after René Descartes, is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It's a foundational graphing system where every point is expressed as an ordered pair \(x, y\).

The plane is divided into four sections called quadrants:

• First Quadrant: Both \(x\) and \(y\) are positive.
• Second Quadrant: \(x\) is negative, and \(y\) is positive.
• Third Quadrant: Both \(x\) and \(y\) are negative.
• Fourth Quadrant: \(x\) is positive, and \(y\) is negative.

Given the example point (-1,1), it's situated in the second quadrant. Here, the x-coordinate is negative and the y-coordinate is positive, depicting that we go left on the x-axis and up on the y-axis from the origin. This coordinate system is pivotal in analytical geometry, facilitating the representation and analysis of geometric figures.

Angle Conversion

Angle conversion is essential when transitioning between rectangular and polar coordinates. Polar coordinates typically include a radius \(r\) and an angle \(\theta\). The process involves calculating these values from the given \(x\) and \(y\) in the Cartesian system.

To obtain the radius \(r\), use the formula:\[r = \sqrt{x^2 + y^2}\]This computes the distance from the point to the origin. For the given point (-1,1), \(r\) becomes \((\sqrt{2})\).

Determining the angle \(\theta\) involves the arctangent function:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]Due to the nature of angle positioning in different quadrants, adjustments are necessary. For instance, a point in the second quadrant like (-1,1) requires adding \(180^\circ\) to account for its true position, resulting in an angle of \(135^\circ\). This understanding is crucial for accurately representing points in polar coordinates, offering a distinct perspective on their position within the plane.