Problem 50 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-2, \frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

(1, -\sqrt{3})

Step 1: Understand the Relationship

Polar coordinates \(r, \theta\) can be converted to rectangular coordinates (x, y) using the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Here, \(r = -2\) and \(\theta = \frac{2 \pi}{3}\).

Step 2: Calculate x-coordinate

Use the formula \(x = r \cos(\theta)\). Substitute the given values: \(x = -2 \cos(\frac{2 \pi}{3})\). Calculate \(\cos(\frac{2 \pi}{3})\), which is \(-\frac{1}{2}\). Therefore, \(x = -2 \times -\frac{1}{2} = 1\).

Step 3: Calculate y-coordinate

Use the formula \(y = r \sin(\theta)\). Substitute the given values: \(y = -2 \sin(\frac{2 \pi}{3})\). Calculate \(\sin(\frac{2 \pi}{3})\), which is \(\frac{\sqrt{3}}{2}\). Therefore, \(y = -2 \times \frac{\sqrt{3}}{2} = -\sqrt{3}\).

Step 4: Combine the Coordinates

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

Key Concepts

polar coordinates

Polar coordinates are a way of locating a point in a plane using a radius (r) and an angle (\theta). Instead of specifying the horizontal (x) and vertical (y) measurements from a reference point (as in rectangular coordinates), polar coordinates give the distance from a reference point (or origin) and the angle from a reference direction (usually the positive x-axis). For example, the polar coordinates \((-2, \frac{2 \pi}{3})\) mean the point is 2 units away from the origin along an angle of \(\frac{2 \pi}{3}\) radians from the positive x-axis.
Using polar coordinates, you can locate a point more easily if it's relevant how far and at what angle the point lies from a central point, such as in navigation or physics problems.

rectangular coordinates

Rectangular coordinates (or Cartesian coordinates) are a standard way to represent points in a 2D plane using an (x, y) format. Each point is described by its horizontal distance (x-coordinate) and vertical distance (y-coordinate) from a fixed reference point (origin). This system sets up a grid where you can easily plot and figure out distances and angles.

• For instance, the point (1, -√3) means it's 1 unit to the right of the origin and √3 units down.
• This approach is highly useful in mathematics, engineering, and many graphing applications.

Rectangular coordinates are sometimes easier to work with when dealing with equations of lines, curves, and other geometric figures.

trigonometric functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions include sine (sin), cosine (cos), tangent (tan), and others. They are fundamental in converting between polar and rectangular coordinates because they provide a relationship between an angle and a ratio of two sides in a right triangle.
In our problem, we use:

• Cosine (\( \cos \theta \)): This function relates the adjacent side to the hypotenuse of a right triangle formed by the x-axis and the point.
• Sine (\( \sin \theta \)): This function relates the opposite side to the hypotenuse in that same triangle.

These trigonometric functions help us find the exact (x, y) positions from the polar coordinates.

coordinate conversion

Coordinate conversion is the process of changing a set of coordinates from one format to another. In mathematics, physics, and engineering, it's often necessary to convert between polar coordinates and rectangular coordinates to solve problems.
For polar to rectangular conversion, we use trigonometric functions:

• The x-coordinate is found using the formula \(x = r \cos(\theta)\).
• The y-coordinate is determined using the formula \(y = r \sin(\theta)\).

To illustrate:

1. Given \(r = -2\) and \(\theta = \frac{2\pi}{3}\), calculate \(\cos(\frac{2\pi}{3})\), which is \(-\frac{1}{2}\).
2. Then multiply: \-2 * -\frac{1}{2} = 1\)
3. Next, calculate \(\textbf{y}\ by finding\ \(\sin(\frac{2\pi}{3})\), which is \(\frac{\sqrt{3}}{2}\)
4. Finally, multiply: \-2 * \(\frac{\sqrt{3}}{2} = -\sqrt{3}\).

Therefore, the rectangular coordinates are (1, -√3). Conversion accuracy ensures precise location representation in different coordinate systems.