Problem 36 - Precalculus Mathematics for Calculus

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(3 \sqrt{3},-3)$$

Step-by-Step Solution

Answer

The polar coordinates are \((6, \frac{11\pi}{6})\).

Step 1: Find the radius, r

The radius in polar coordinates is given by the formula \( r = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the rectangular coordinates. Given \( x = 3\sqrt{3} \) and \( y = -3 \), we substitute these values into the formula:\[r = \sqrt{(3\sqrt{3})^2 + (-3)^2} = \sqrt{27 + 9} = \sqrt{36} = 6\]

Step 2: Find the angle, θ

The angle \( \theta \) can be calculated using \( \tan \theta = \frac{y}{x} \). Substitute in the values for \( x \) and \( y \):\[\tan \theta = \frac{-3}{3\sqrt{3}} = -\frac{1}{\sqrt{3}}\]This corresponds to an angle of \( \theta = -\frac{\pi}{6} \). Since the angle must be between \( 0 \) and \( 2\pi \), we convert it to a positive angle by adding \( 2\pi \):\[\theta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}\]

Step 3: Compile the polar coordinates

Now that we have both \( r \) and \( \theta \), we can compile the polar coordinates. The coordinates in polar form are \((r, \theta)\). Thus, the polar coordinates are:\[(6, \frac{11\pi}{6})\]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are defined by a pair of numerical values: \(x, y\). These values represent a point's position in a two-dimensional plane. In our case, we are given the rectangular coordinates \(3\sqrt{3}, -3\). The \(x\) value, also called the abscissa, represents the horizontal distance from the origin, while the \(y\) value, known as the ordinate, represents the vertical distance from the origin.
Understanding rectangular coordinates is crucial because they provide a straightforward way to identify a position in space. When moving to polar coordinates, we translate this information into a distance from the origin and an angle from the positive x-axis. This transformation is essential in scenarios that involve circular and rotational contexts, such as in physics and engineering.

Coordinate Transformation

Coordinate transformation is the process of converting coordinates from one system to another. In this exercise, we are transforming from rectangular (Cartesian) coordinates to polar coordinates.
This transformation involves two main steps:

• Calculating the radius \(r\).
• Finding the angle \(\theta\).

Polar coordinates are defined by \(r\) and \(\theta\), where \(r\) is the distance from the origin to the point, and \(\theta\) is the angle measured from the positive x-axis. The coordinate transformation thus involves rearticulating the location of a point based on these circular arguments. It's a handy method when you want to describe locations in contexts involving rotation and circular motion.

Angle Calculation

To find the angle \(\theta\), we use the arctangent function, which gives us the angle that corresponds to the given tangent. The relationship is expressed as \( \tan \theta = \frac{y}{x} \).
Using the given values, we compute:
\[ \tan \theta = \frac{-3}{3\sqrt{3}} = -\frac{1}{\sqrt{3}} \]
The \(\theta\) we find initially is \(-\frac{\pi}{6}\). However, polar coordinates require \(\theta\) to be expressed within the range of \(0\) and \(2\pi\). Therefore, we convert to a positive angle by adding \(2\pi\):
\[ \theta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6} \]Remember that choosing the correct quadrant for \(\theta\) is key, as it affects both the direction and nature of the point's location. This careful calculation ensures accurate mapping from rectangular to polar coordinates.

Radius Computation

The radius \(r\) is calculated using the Pythagorean theorem. This is done by taking the square root of the sum of the squares of the \(x\) and \(y\) coordinates.
For our given point, the formula looks like:
\[ r = \sqrt{(3\sqrt{3})^2 + (-3)^2} \]
Breaking this down step-by-step:

• Compute \( (3\sqrt{3})^2 = 27 \).
• Compute \( (-3)^2 = 9 \).
• Sum them to get \( 27 + 9 = 36 \).
• Finally, find the square root \( \sqrt{36} = 6 \).

The radius \(r\) describes how far the point is from the origin. This calculation is a core part of moving to polar coordinates as it shifts the focus from a direct axis-based approach to one centered on distance and direction.