Problem 16 - Contemporary Precalculus

Question

Convert the polar coordinates to rectangular coordinates. $$(2,0)$$

Step-by-Step Solution

Answer

Question: Convert the polar coordinates (2, 0) to rectangular coordinates. Answer: The rectangular coordinates equivalent to the polar coordinates (2, 0) are (2, 0).

Step 1: Identify the polar coordinates

We are given the polar coordinates \((2, 0)\). The radial distance from the origin, \(r\), is 2, and the angle counterclockwise from the positive x-axis, \(θ\), is 0.

Step 2: Convert the radial distance and angle to rectangular coordinates

Using the relationships between polar and rectangular coordinates, compute the rectangular coordinates \((x, y)\): $$x = r\cos(θ) = 2\cos(0) = 2(1) = 2$$ $$y = r\sin(θ) = 2\sin(0) = 2(0) = 0$$

Step 3: Write the final answer

The rectangular coordinates equivalent to the polar coordinates \((2, 0)\) are \((2, 0)\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing a point in space using a distance and an angle. Instead of using x and y like in rectangular coordinates, polar coordinates use:

• r: The distance from the origin (center) to the point.
• θ: The angle made with the positive x-axis, measured counterclockwise.

For example, the polar coordinates \((2, 0)\) tell us that the point is 2 units away from the origin and directly along the positive x-axis. This makes it particularly simple to locate the point in the polar coordinate system. Polar coordinates are especially useful in contexts like circular motion or waves where angles and distances offer a more natural way to describe the point's location.
Understanding the way polar coordinates work helps in converting them to other forms like rectangular coordinates, offering different insights for different problems.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, describe a point using an x and a y value. These coordinates form a grid where a point is specified by its horizontal and vertical distances from the origin. This system divides the plane into four quadrants:

• x: Horizontal distance from the origin.
• y: Vertical distance from the origin.

In the example given, the rectangular coordinates equivalent to the polar coordinates \((2, 0)\) are \((2, 0)\). In this particular case, the conversion results in the same values, but this is not generally true for all polar coordinates. The conversion involves using trigonometric functions:

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

This makes rectangular coordinates straightforward to use for problems involving linear motion or grid-based systems.

Coordinate Systems

Coordinate systems, essential tools in mathematics and physics, are methods to specify the position of a point in space. The two most common systems are polar and rectangular coordinates.
Polar coordinates are better suited for circular or rotational systems where angles and radii naturally describe the position.

• Useful for circles, spirals, and any rotational system.

Rectangular coordinates, in contrast, work best for grid-like, linear systems, where movement along straight lines is involved.

• Ideal for linear translations, functions, and geometry.

Conversion between the systems allows flexibility in problem-solving, letting you choose the most convenient form based on the context. Understanding these systems allows one to switch concepts easily and evaluate the situation accurately, which is essential in many scientific and engineering applications.