Problem 67 - Precalculus

Question

Use a graphing utility to find one set of polar coordinates of the point given in rectangular coordinates. $$\left(\frac{5}{2}, \frac{4}{3}\right)$$

Step-by-Step Solution

Answer

The polar coordinates of the rectangular point \((\frac{5}{2}, \frac{4}{3})\) are \((\frac{17}{6}, \arctan(\frac{8}{15}))\).

Step 1: Compute the radial coordinate r

We find the radial coordinate r by transforming the rectangular coordinates into polar form. The \(r\) value is calculated using the formula \(r = \sqrt{x^2 + y^2}\). In this instance, \(x = \frac{5}{2}\) and \(y = \frac{4}{3}\). Thus, \(r = \sqrt{(\frac{5}{2})^2 + (\frac{4}{3})^2} = \sqrt{\frac{25}{4} + \frac{16}{9}} = \sqrt{\frac{225 + 64}{36}} = \sqrt{\frac{289}{36}} = \frac{17}{6}\)

Step 2: Compute the Angular Coordinate

After determining the radial coordinate, the next step is to figure out the angular coordinate \(\theta\). This value provides the angle between the positive x-axis and the line connecting the origin to the given point in the plane. The formula is \(\theta = \arctan(\frac{y}{x})\), which in this case is \(\theta = \arctan(\frac{4}{3}/\frac{5}{2}) = \arctan(\frac{8}{15})\). This is in radians. To convert into degrees (if needed), we multiply by \(\frac{180}{\pi}\)

Step 3: Finalize the Polar Coordinates

Once we've computed the radial and angular coordinates, our polar coordinates for the given point are \((r, \theta)\). After using a graphing utility to verify, the point \((\frac{5}{2}, \frac{4}{3})\) in rectangular coordinates is \((\frac{17}{6}, \arctan(\frac{8}{15}))\) in polar coordinates. You can see the transformation on your graphing utility. Notice that there can be multiple sets of polar coordinates corresponding to this point due to periodicity in the case we add or subtract multiples of \(2\pi\) to/from \(\theta\), but we just wanted one set.

Key Concepts

Rectangular to Polar Conversion

Converting coordinates from rectangular (also known as Cartesian) to polar form allows us to express points on a plane using a different perspective. Instead of describing a position in terms of horizontal and vertical distances (x and y), polar coordinates use a combination of a radial distance from the origin (r) and an angular displacement from the positive x-axis (θ).

In practice, converting a point like \(\left(\frac{5}{2}, \frac{4}{3}\right)\) from rectangular to polar coordinates entails finding the correct values for r and θ. This is particularly useful in situations where circular motion or symmetry is involved, as it often simplifies calculations and visual understanding.

Radial Coordinate Calculation

The radial coordinate, denoted as r, represents the distance from the origin to the point and is the foundation of the polar coordinate system. To calculate r, we use the Pythagorean Theorem because a point in rectangular coordinates essentially forms a right triangle with the x and y axes serving as legs.

Here's how to calculate r:

• Identify the x (horizontal) and y (vertical) coordinates.
• Apply the formula \(r = \sqrt{x^2 + y^2}\).
• For our example, we have \(x = \frac{5}{2}\) and \(y = \frac{4}{3}\), so \(r = \sqrt{\left(\frac{5}{2}\right)^2 + \left(\frac{4}{3}\right)^2} = \frac{17}{6}\).

An accurate calculation of r is crucial as it quantifies how far away a point is from the center of the polar coordinate system.

Angular Coordinate Calculation

The angular coordinate, θ, defines the direction of the radius with reference to the positive x-axis. The angle is measured in radians, with one full revolution equal to \(2\pi\) radians, or 360 degrees. To calculate θ, you will need the following steps:

To find θ:

• Take the y (vertical) and x (horizontal) coordinates.
• Use the formula \(θ = \arctan(\frac{y}{x})\).
• For the point given, \(\theta; = \arctan(\frac{4}{3}/\frac{5}{2}) = \arctan(\frac{8}{15})\).
• If desired, convert θ from radians to degrees with the conversion \(θ \times \frac{180}{\pi}\).

Understanding how to calculate θ is essential because it specifies the exact location of the point around the circle, enabling the complete and precise description of the point's position in polar coordinates.