Problem 40 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Components of Polar Coordinates

Polar coordinates are expressed in the form \(r, \theta\). Here, \(r\) represents the radial distance from the origin to the point, and \(\theta\) represents the angle measured from the positive x-axis to the line segment connecting the origin with the point.

Step 2: Determine the Radial Distance

The radial distance \(r\) is calculated using the formula \[r = \sqrt{x^2 + y^2}\]. Substitute \(x = 0\) and \(y = -\sqrt{3}\) into the formula:\[r = \sqrt{0^2 + (-\sqrt{3})^2} = \sqrt{3}\].Thus, \(r = \sqrt{3}\).

Step 3: Calculate the Angle \(\theta\)

The angle \(\theta\) is calculated using the arctangent function: \[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]. Substitute \(x = 0\) and \(y = -\sqrt{3}\) to find:\[\tan^{-1}\left(\frac{-\sqrt{3}}{0}\right)\].Since \(x = 0\), the point lies on the negative y-axis, and thus \(\theta = \frac{3\pi}{2}\) because it is exactly downward.

Step 4: Combine \(r\) and \(\theta\) into Polar Coordinates

Now, we combine the values obtained for \(r\) and \(\theta\) into the polar coordinate form. Thus, the polar coordinates for the point \( (0, -\sqrt{3})\) are: \[ (\sqrt{3}, \frac{3\pi}{2}) \].

Key Concepts

Rectangular to Polar Conversion

Converting between coordinate systems is essential in mathematics and physics. Rectangular coordinates, also known as Cartesian coordinates, use the format \(x, y\), representing a grid-based location in space.
In contrast, polar coordinates use a radial and angular system \(r, \theta\), making them particularly useful in scenarios involving circular or rotational contexts.
To convert rectangular coordinates to polar coordinates:

• First, identify the x and y values given in the rectangular coordinates.
• Determine the radial distance \(r\).
• Calculate the angle \(\theta\).

Here, converting the point \( (0, -\sqrt{3}) \) to polar form involves finding both of these components and expressing them as a pair, \( (r, \theta) \). This method provides a full understanding of the point in a circular coordinate system.

Radial Distance Calculation

The radial distance, often denoted as \(r\), is a fundamental part of polar coordinates and determines how far the point is from the origin.
This distance can be thought of as the hypotenuse of a right triangle formed by the x and y coordinates.
The formula for the radial distance is: \[ r = \sqrt{x^2 + y^2} \]To calculate, substitute the x and y components into the equation:

• For \(x = 0\), and \(y = -\sqrt{3}\), substitute these to find \(r\).
• Calculate: \( r = \sqrt{0^2 + (-\sqrt{3})^2} = \sqrt{3} \).

Thus, the radial distance \(r\) is \(\sqrt{3}\). This measure assures how far the point is from the origin, independent of the angle.

Angle Measurement in Polar Coordinates

Understanding angle measurement is crucial for working with polar coordinates. The angle \(\theta\) represents the direction of the radial line from the origin, measured counterclockwise from the positive x-axis.
When finding the angle, especially under special cases like when \(x = 0\), it's important to recognize the line's position relative to the axes.
For the coordinates \( (0, -\sqrt{3}) \):

• The line lies on the negative y-axis.
• Thus \(\theta = \frac{3\pi}{2}\), which corresponds to 270° in angle measure.

Working through special cases like these, where the division by zero might occur in the standard \(\tan^{-1}(\frac{y}{x})\) calculation, ensures accurate angle representation without needing complex calculations. This pragmatic understanding of angles helps to fully visualize points in physical space using the polar system.