Problem 32 - Algebra and Trigonometry

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (\sqrt{3},-5 \pi / 3) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left( \frac{\sqrt{3}}{2}, \frac{3}{2} \right)\).

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis. Here, \(r = \sqrt{3}\) and \(\theta = -\frac{5\pi}{3}\).

Step 2: Use the Conversion Formulas

The conversion from polar coordinates to rectangular coordinates \((x, y)\) involves the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\). We will use these formulas with our given \(r\) and \(\theta\).

Step 3: Calculate the \(x\)-Coordinate

Calculate \(x\) using the formula: \(x = \sqrt{3} \cos \left(-\frac{5\pi}{3}\right)\). The angle \(-\frac{5\pi}{3}\) is equivalent to \(\frac{\pi}{3}\) since angles are periodic with period \(2\pi\). Thus, \(x = \sqrt{3} \cos \frac{\pi}{3} = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2}\).

Step 4: Calculate the \(y\)-Coordinate

Calculate \(y\) using the formula: \(y = \sqrt{3} \sin \left(-\frac{5\pi}{3}\right)\). The angle \(-\frac{5\pi}{3}\) is equivalent to \(\frac{\pi}{3}\). Thus, \(y = \sqrt{3} \sin \frac{\pi}{3} = \sqrt{3} \times \frac{\sqrt{3}}{2} = \frac{3}{2}\).

Step 5: State the Rectangular Coordinates

The rectangular coordinates \((x, y)\) are \(\left( \frac{\sqrt{3}}{2}, \frac{3}{2}\right)\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of defining a point in a plane using two values: a distance and an angle.

• Radial Distance (r): This represents how far the point is from the origin. A positive value indicates distance, while a negative value flips the point across the origin.
• Angle (\(\theta\)): This is measured from the positive x-axis towards the point in a counter-clockwise direction. Sometimes angles can be negative, indicating a clockwise direction.

For the given problem, we have polar coordinates \((\sqrt{3}, -\frac{5\pi}{3})\). Here, \(r = \sqrt{3}\) determines the distance, and \(\theta = -\frac{5\pi}{3}\) represents an angle that needs conversion for a simpler understanding.

Rectangular Coordinates

Rectangular coordinates offer another way to pinpoint locations on a plane, using a familiar x-y coordinate grid.

• X-coordinate (x): Represents the horizontal distance from the y-axis.
• Y-coordinate (y): Represents the vertical distance from the x-axis.

Each point in rectangular coordinates is represented as \((x, y)\). The aim is to convert polar coordinates to rectangular ones, which is useful in various applications like navigation and engineering.

Angle Conversion

Converting angles is crucial to finding accurate rectangular coordinates. Angles can be expressed in many forms, commonly in degrees or radians.

• Periodicity: Angles repeat every \(2\pi\) radians or 360 degrees. This means an angle like \(-\frac{5\pi}{3}\) can be adjusted by adding \(2\pi\) to bring it into a positive range without changing the point its location.

To convert \(-\frac{5\pi}{3}\) to a positive angle, we add \(2\pi\), resulting in \(\theta = \frac{\pi}{3}\). This adjusted angle can now be effectively used in trigonometric calculations.

Trigonometric Functions

Trigonometric functions form the foundation of converting polar to rectangular coordinates. Specifically, cosine and sine are used:

• Cosine (\(\cos\)): relates to the x-coordinate. It helps determine how far along or away from the x-axis a point is.
• Sine (\(\sin\)): relates to the y-coordinate. It helps determine how far above or below the x-axis a point is.

Conversion formulas are:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

Using \(r = \sqrt{3}\) and the converted \(\theta = \frac{\pi}{3}\):- The \(x\)-coordinate is calculated as: \(x = \sqrt{3} \cos \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).- The \(y\)-coordinate is calculated as: \(y = \sqrt{3} \sin \frac{\pi}{3} = \frac{3}{2}\).Thus, the rectangular coordinates are found by these trigonometric applications.