Problem 53 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (24,-7) $$

Step-by-Step Solution

Answer

The polar coordinates are \((25, 5.999)\).

Step 1: Find the Radius

The radius in polar coordinates is represented by \( r \) and is calculated using the formula \( r = \sqrt{x^2 + y^2} \) where \( x \) and \( y \) are the rectangular coordinates.In this case, \( x = 24 \) and \( y = -7 \).Compute the radius:\[ r = \sqrt{24^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

Step 2: Calculate the Angle

The angle \( \theta \) in polar coordinates is found using the arctangent function: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).For this point, \( x = 24 \) and \( y = -7 \). Therefore,\[ \theta = \tan^{-1}\left(\frac{-7}{24}\right) \]Using a calculator, this value is approximately \( \theta = -0.284 \) radians. Since angles in polar coordinates must be between \( 0 \) and \( 2\pi \), add \( 2\pi \) to \( -0.284 \) to find \( \theta \) in the correct range:\[ \theta = 2\pi - 0.284 \approx 5.999 \text{ radians} \]

Step 3: Write the Polar Coordinates

Combine the values of \( r \) and \( \theta \) to express the point in polar coordinates:The polar coordinates are \( (25, 5.999) \).

Key Concepts

Rectangular Coordinates

When we talk about rectangular coordinates, we are referring to a pair of numbers that determine the position of a point in a flat plane, also known as a Cartesian coordinate plane. This is the most common system you learn in math, typically shown as

• The first number being the 'x-coordinate', which tells you how far the point is from the vertical axis (also called the y-axis).
• The second number being the 'y-coordinate', which indicates how far the point is from the horizontal axis (likewise called the x-axis).

Rectangular coordinates are typically written as \((x, y)\). For example, the point \((24, -7)\) describes a location 24 units to the right of the origin along the x-axis and 7 units down along the y-axis. They are fundamental in graphing and are used extensively in sciences and engineering. These coordinates make it easy to plot points, shapes, or even visualize how equations behave when graphed.

Polar Coordinates

Polar coordinates offer a different way of plotting points, focusing on the distance from a central point and the angle from a reference direction. Unlike rectangular coordinates, polar coordinates are expressed with \((r, \theta)\), where:

• \(r\) represents the radius, or the distance from the central point, often referred to as the origin.
• \(\theta\) is the angle measured in radians from the positive x-axis, moving counterclockwise around the plane.

To convert a rectangular coordinate to a polar coordinate, you first need to calculate \(r\) using the formula \(r = \sqrt{x^2 + y^2}\).
The angle \(\theta\) can be found using the arctangent function, described in the next section. Polar coordinates are useful in systems where angles and distances from a central point are more intuitive or easier to use, such as navigation and in cases dealing with circular motion.

Arctangent Function

Angles in polar coordinates are often measured using the arctangent function, also known as the inverse tangent function. This function, denoted as \(\tan^{-1}\) or \(\arctan\), is used to find an angle whose tangent is a given number.
The formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\) is applied when converting from rectangular to polar coordinates to determine the angle \(\theta\) that a point makes with the positive x-axis.
Some things to remember:

• The range of \(\tan^{-1}\) is \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) which means it returns values that range only within this interval.
• If the computed angle is negative, adding \(2\pi\) will adjust it so that it lies within the standard polar coordinate range of \(0\) to \(2\pi\).

This function is vital in trigonometry and is frequently used in situations requiring calculations of angles from slopes, such as in engineering and computer graphics.