Problem 51 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (-5,-12) $$

Step-by-Step Solution

Answer

The polar coordinates are approximately (13, 4.29).

Step 1: Determine the Value of r

To convert from rectangular to polar coordinates, we need to determine the radius \( r \), which is the distance from the origin to the point \((-5,-12)\). The formula to calculate \( r \) in polar coordinates is: \[ r = \sqrt{x^2 + y^2} \] Substitute the given values \( x = -5 \) and \( y = -12 \):\[ r = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] Therefore, \( r = 13 \).

Step 2: Determine the Angle θ

Next, we find the angle \( \theta \) using the arctangent function \( \tan^{-1} \), where \( \tan \theta = \frac{y}{x} \):\[ \theta = \tan^{-1}\left(\frac{-12}{-5}\right) \]This simplifies to \[ \theta = \tan^{-1}(2.4) \].On evaluating, we consider that our point lies in the third quadrant (since both coordinates are negative). The angle in radians that makes \( \tan \theta = 2.4 \) is approximately \( 1.15 \). Thus, to adjust for the third quadrant:\[ \theta_{polar} = \theta + \pi = 1.15 + 3.14 = 4.29 \] Hence, \( \theta \approx 4.29 \) radians.

Step 3: Express Polar Coordinates

Now that we have both the radius and the angle, the polar coordinates \((r, \theta)\) can be written as:\[ (13, 4.29) \]

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, denote a point's location in a plane using two values. These values, commonly represented as \((x, y)\), specify horizontal and vertical distances from the origin \((0, 0)\).

• The first number in the pair, \(x\), indicates the position relative to the vertical axis. Positive values of \(x\) mean the point is right of the origin, while negative values indicate it is to the left.
• The second number, \(y\), determines how far the point is from the horizontal axis. Positive \(y\) values place it above the origin, and negative values place it below.

This coordinate system is integral to converting points into different forms, such as polar coordinates. Understanding this system clarifies the relationships between point locations and various calculations, like finding the distance or determining which quadrant a point lies in.

Applying the Distance Formula

The distance formula is an essential tool for calculating the straight-line distance between two points in a plane. In this context, it is used to determine the radius \(r\) when converting rectangular coordinates to polar coordinates.The formula is:\[ r = \sqrt{x^2 + y^2}\]Here's how it works:- Each coordinate \((x, y)\) is squared, eliminating negative signs and reflecting how far the point is horizontally and vertically from the origin.- These squares add together, and the square root of the sum gives the distance, \(r\).For example, consider the point \((-5, -12)\):

• Square each coordinate: \((-5)^2 = 25\) and \((-12)^2 = 144\)
• Add the squares: \(25 + 144 = 169\)
• Take the square root: \(\sqrt{169} = 13\)

Thus, the distance from the origin, \(r\), is 13, which represents the radius in polar coordinates.

Identifying Quadrants

The plane in rectangular coordinates is divided into four quadrants. These quadrants help in identifying the position of a point and are numbered counterclockwise:

• Quadrant I: \(x > 0\) and \(y > 0\)
• Quadrant II: \(x < 0\) and \(y > 0\)
• Quadrant III: \(x < 0\) and \(y < 0\)
• Quadrant IV: \(x > 0\) and \(y < 0\)

For a point like \((-5, -12)\), both \(x\) and \(y\) are negative, placing it in Quadrant III. Understanding which quadrant a point belongs to is crucial when converting to polar coordinates. This information ensures that the angle \(\theta\) is correctly adjusted. For Quadrant III, this includes adding \(\pi\) (or 180 degrees) to the calculated angle \(\theta\), ensuring the direction from the positive x-axis is rightly represented in the polar form. In this way, the conversion maintains an accurate reflection of the point's position on the plane.