Problem 27 - Precalculus

Question

Convert each point to exact rectangular coordinates. $$\left(2,240^{\circ}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Understand Polar Coordinates

Polar coordinates are given as \((r, \theta)\), where \(r\) is the radius or the distance from the origin, and \(\theta\) is the angle measured from the positive x-axis. Our point is \((2, 240^{\circ})\).

Step 2: Use Formulas for Conversion

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

Step 3: Calculate the x-coordinate

Substitute \(r = 2\) and \(\theta = 240^{\circ}\) into the formula for \(x\): \[x = 2 \cos(240^{\circ})\]. \(\cos(240^{\circ}) = -\frac{1}{2}\), so \(x = 2 \times -\frac{1}{2} = -1\).

Step 4: Calculate the y-coordinate

Substitute \(r = 2\) and \(\theta = 240^{\circ}\) into the formula for \(y\): \[y = 2 \sin(240^{\circ})\].\(\sin(240^{\circ}) = -\frac{\sqrt{3}}{2}\), so \(y = 2 \times -\frac{\sqrt{3}}{2} = -\sqrt{3}\).

Step 5: Write the Final Answer

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

Key Concepts

Polar Coordinates

Polar coordinates are a way of expressing points in a plane using a combination of a radius and an angle. Instead of referring to a point based on its horizontal and vertical offsets from the origin (as you would with rectangular coordinates), you use:

• Radius (\(r\)): This is the distance from the origin (0,0) to the point in question. In our example, the radius is 2.
• Angle (\(\theta\)): This is the angle measured in degrees or radians from the positive x-axis to the line connecting the origin and the point. In our case, the angle is \(240^{\circ}\).

To visualize polar coordinates, imagine a circle centered at the origin. The radius tells you how far out from the center to go, and the angle tells you in which direction to go. This system is particularly useful for solving problems involving circular or rotational symmetry.

Rectangular Coordinates

Rectangular coordinates, sometimes called Cartesian coordinates, refer to points defined by horizontal and vertical units away from an origin. These are expressed as

• x-coordinate: Indicates the horizontal distance from the y-axis. In our conversion problem, this is calculated using the formula \(x = r \cos(\theta)\).
• y-coordinate: Represents the vertical distance from the x-axis. This is determined using the formula \(y = r \sin(\theta)\).

The primary benefit of rectangular coordinates is their ease when it comes to performing algebraic operations, especially when dealing with straight lines or rectangular objects. When you have coordinates such as \(-1\) and \(-\sqrt{3}\), it indicates a left and downward shift from the origin in a standard 2D plane.

Coordinate System Conversion

Converting between polar and rectangular coordinates involves straightforward trigonometric operations. Here is how it works:

• Given polar coordinates \((r, \theta)\), calculate the rectangular coordinates \((x, y)\) using the conversion formulas:
• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

For the point \((2, 240^{\circ})\), we plugged in the radius and angle into the conversion formulas:

• We determined \(x = 2 \cos(240^{\circ}) = -1\).
• Then, \(y = 2 \sin(240^{\circ}) = -\sqrt{3}\).

This process involves using trigonometric values for common angles, which you can often find on a unit circle. Through these steps, one can effectively transition between coordinate systems, leveraging each system's respective strengths for problem-solving and analysis in geometry and physics.