Problem 18 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(2, \frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((1, \sqrt{3})\).

Step 1: Identify Polar Coordinates

The given polar coordinates are \((r, \theta) = (2, \frac{\pi}{3})\). Here, \(r = 2\) is the radius and \(\theta = \frac{\pi}{3}\) is the angle.

Step 2: Use Formulas for Rectangular Coordinates

To convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas: \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\).

Step 3: Calculate x-coordinate

Substitute \(r = 2\) and \(\theta = \frac{\pi}{3}\) into the formula for \(x\):\[x = 2 \cdot \cos\left(\frac{\pi}{3}\right) = 2 \cdot \frac{1}{2} = 1\]

Step 4: Calculate y-coordinate

Substitute \(r = 2\) and \(\theta = \frac{\pi}{3}\) into the formula for \(y\):\[y = 2 \cdot \sin\left(\frac{\pi}{3}\right) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}\]

Step 5: Write Rectangular Coordinates

The rectangular coordinates are \((x, y) = (1, \sqrt{3})\).

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way to locate points in a plane. Instead of using conventional horizontal and vertical lines, as in the Cartesian system, polar coordinates use a distance and an angle. The distance, denoted as \(r\), measures how far the point is from a fixed origin, often called the pole. Meanwhile, the angle, \(\theta\), represents the direction from the pole, typically measured in radians. Radians are preferred as they relate directly to the circle's radius and circumference. The angle \(\theta\) is taken from the positive x-axis, rotating counter-clockwise.
This system is especially helpful in situations involving circular or rotational symmetry, like waves or circular motion. Key benefits include simplicity in representing curved paths, since curves such as circles are easily defined with polar coordinates compared to rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are perhaps the most familiar to math students. This system uses two numbers, \(x\) and \(y\), to locate a point in a plane. Here, the horizontal distance from the origin is given by \(x\), whereas \(y\) provides the vertical distance.
This system finds widespread use due to its straightforward nature and direct applications in various fields like engineering, computer graphics, and physics. It allows for easy calculation when working with straight lines and is the foundation for many mathematical operations.

• Natural fit for equations representing lines and curves.
• Useful in calculating areas and volumes.

Understanding rectangular coordinates is crucial because they serve as a bridge to more complex coordinate systems.

Trigonometric Functions

Trigonometric functions such as \(\cos\) and \(\sin\) play an integral role in converting between coordinate systems. These functions relate angles to side ratios in right triangles, which is why they are pivotal in conversions.
In polar to rectangular conversion, the \(x\)-coordinate is found using the formula \(x = r \cdot \cos(\theta)\), and the \(y\)-coordinate using \(y = r \cdot \sin(\theta)\). This linkage arises because the circular nature of trigonometric functions perfectly aligns with the polar system.

• \(\cos(\theta)\) provides the adjacent side's ratio (horizontal projection).
• \(\sin(\theta)\) provides the opposite side's ratio (vertical projection).

These functions simplify and clarify how distances and angles relate, making the transformation between coordinate systems an achievable task.

Coordinate Conversion

Coordinate conversion is the process of translating points between different coordinate systems without losing any positional information. It is like translating languages; the meaning remains, but the expression changes.
In the context of polar to rectangular conversion, this involves using the trigonometric functions \(\cos\) and \(\sin\) to transform polar coordinates \((r, \theta)\) into rectangular coordinates \((x, y)\). The formulas \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\) are essential tools in this task.
This process is crucial in many applications, such as navigation, where different forms of data presentation enhance understanding and calculations. Whether for plotting paths on a GPS or modeling physics problems, converting between systems ensures the versatility needed to approach problems from multiple angles. Understanding this allows one to fluently switch and apply the advantages of each system optimally.