Problem 26 - Algebra and Trigonometry

Question

A point is graphed in polar form. Find its rectangular coordinates. $$ (6,2 \pi / 3) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (-3, 3√3).

Step 1: Understanding Polar Coordinates

Polar coordinates are given in the form \(r, \, \theta\), where \(r\) is the radius (distance from origin), and \(\theta\) is the angle from the positive x-axis measured counterclockwise. In this problem, \(r = 6\) and \(\theta = \frac{2\pi}{3}\).

Step 2: Convert Angle to Radians

The angle \(\frac{2\pi}{3}\) is already in radians, which is necessary for conversion to rectangular coordinates, so we do not need to change the angle format.

Step 3: Finding the x-coordinate

The x-coordinate in rectangular form can be found using the formula \(x = r \cos(\theta)\). Substitute \(r = 6\) and \(\theta = \frac{2\pi}{3}\) into the formula: \[x = 6 \cos\left(\frac{2\pi}{3}\right)\]. Since \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\), we have \[x = 6 \times -\frac{1}{2} = -3\].

Step 4: Finding the y-coordinate

The y-coordinate in rectangular form can be found using the formula \(y = r \sin(\theta)\). Substitute \(r = 6\) and \(\theta = \frac{2\pi}{3}\) into the formula: \[y = 6 \sin\left(\frac{2\pi}{3}\right)\]. Since \(\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\), we have \[y = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3}\].

Step 5: Conclusion

The rectangular coordinates of the given polar point \(6, \frac{2\pi}{3}\) are \((-3, \, 3\sqrt{3})\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Think of it like using directions to locate treasure on a map. Instead of 'x' and 'y' like in a rectangular system, polar coordinates use 'r' and 'θ'. Here is what these mean:

• Distance (r): This is how far the point is from the origin (the center point of the graph).
• Angle (θ): The angle is measured from the positive x-axis in a counterclockwise direction.

In the exercise, the polar coordinates given are (6, 2π/3). This means the point is 6 units away from the origin and 2π/3 radians around the circle. Polar coordinates are perfect for circular and rotational systems, making them very useful in fields such as engineering and physics.

When starting with polar coordinates, always remember you're working with circles and angles. It's like plotting a point on a radar screen or determining where a ship is on the sea!

Rectangular Coordinates

Rectangular coordinates, sometimes known as Cartesian coordinates, use two values to locate a point on a plane: 'x' and 'y'. This system is similar to a regular map divided by horizontal and vertical lines. Each point has:

• x-coordinate: This tells you how far to move along the x-axis (horizontally).
• y-coordinate: This shows how far to move along the y-axis (vertically).

Converting from polar to rectangular coordinates involves finding these 'x' and 'y' values using the radius and angle from polar coordinates. Think of this as translating a point from a circular grid to a square one.

In the exercise, the point (6,2π/3) was translated into rectangular coordinates using formulas. The result was x = -3 and y = 3√3. Always remember, rectangular coordinates make measuring straight lines simple, which is why they are widely used in geometry and calculating distances.

Trigonometric Functions

Trigonometric functions like sine (sin) and cosine (cos) play a crucial role in converting polar coordinates to rectangular coordinates. They help bridge the gap between angles and straight lines. Here's how they function in conversion:

• Cosine Function:
  Used to find the x-coordinate. The formula is x = r cos(θ). It calculates the horizontal distance from the origin based on the radius and angle.
• Sine Function:
  Used to find the y-coordinate. The formula is y = r sin(θ). It computes the vertical distance from the origin.

In our exercise, with 'r' as 6 and 'θ' as 2π/3 radians:

• The x-coordinate was found using x = 6 cos(2π/3), which equals -3.
• The y-coordinate was found using y = 6 sin(2π/3), which equals 3√3.

Trigonometric functions simplify handling circular motion and waves effectively. Remembering these relationships helps you understand how curves can be translated into straight lines or vice versa.