Problem 2 - Contemporary Precalculus

Question

Plot the point whose polar coordinates are given. $$(1, \pi / 4)$$

Step-by-Step Solution

Answer

Answer: The Cartesian coordinates of the point are (sqrt(2)/2, sqrt(2)/2).

Step 1: Understand polar coordinates and their relation with Cartesian coordinates

In polar coordinates, a point is described by the distance (radius) from the origin and the angle between the positive x-axis and the line connecting the origin to the point. The polar coordinates are denoted by \((r, \theta)\), where \(r\) is the radius, and \(\theta\) is the angle. To convert polar coordinates to Cartesian coordinates, we use the following relations: $$x = r \cos{\theta}$$ $$y = r \sin{\theta}$$

Step 2: Convert the given polar coordinates to Cartesian coordinates

We are given the polar coordinates, \((1, \pi / 4)\). Thus, the radius \(r = 1\) and the angle \(\theta = \pi / 4\). Using the relations mentioned in step 1, we can find the Cartesian coordinates. $$x = r \cos{\theta} = 1 \cos{(\pi / 4)} = \frac{\sqrt{2}}{2}$$ $$y = r \sin{\theta} = 1 \sin{(\pi / 4)} = \frac{\sqrt{2}}{2}$$ So, the Cartesian coordinates are \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\).

Step 3: Plot the point in Cartesian coordinates

Now that we have the Cartesian coordinates, we can plot the point on a grid. 1. Start at the origin (0,0). 2. Move to the right by \(\frac{\sqrt{2}}{2}\) units on the x-axis. 3. Move up by \(\frac{\sqrt{2}}{2}\) units on the y-axis. 4. Place the point at the location \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). The point with polar coordinates \((1, \pi / 4)\) is now plotted on the grid as Cartesian coordinates \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are one of the most common systems used to describe a point's location in a plane. In this system, every point is specified by two numbers which are usually referred to as

• x: The horizontal coordinate,
• y: The vertical coordinate.

The plane itself is divided by two axes – the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, which has the coordinates (0,0).

When you plot a point using Cartesian coordinates, you start from the origin. You move horizontally along the x-axis by the value of x, and then vertically along the y-axis by the value of y. This system was named after René Descartes, a French mathematician, who was crucial in the development of analytic geometry. It's widely used in mathematics, engineering, physics, and even game development to identify locations in a plane.

Coordinate Conversion

Coordinate conversion is a process of translating values from one coordinate system to another. This is particularly important in fields like navigation, computer graphics, and robotics, where different format systems may apply.

For example, when converting from polar to Cartesian coordinates, we utilize certain mathematical formulas. These formulas help translate a point defined by distance and angle (polar coordinates) into a point defined by horizontal and vertical displacement (Cartesian coordinates).

The key relationships for this conversion are:

• For the x-coordinate: \(x = r \cos{\theta}\)
• For the y-coordinate: \(y = r \sin{\theta}\)

Here, \(r\) is the radial distance from the origin, and \(\theta\) is the angle from the positive x-axis. Coordinate conversion is not only an essential mathematical skill but also deeply rooted in many practical applications.

Polar to Cartesian Transformation

The transformation from polar to Cartesian coordinates is an important mathematical procedure that bridges two different methods of describing points. It allows us to convert a point expressed as radius and angle to x and y coordinates. In our example, the polar coordinates (1, \(\pi/4\)) are transformed to Cartesian.

• The radius \(r = 1\) determines how far the point is from the origin.
• The angle \(\theta = \pi/4\) indicates its direction from the positive x-axis.

Using the relationships for conversion:

• \(x = r \cos{\theta} = 1 \cos{(\pi/4)} = \frac{\sqrt{2}}{2}\)
• \(y = r \sin{\theta} = 1 \sin{(\pi/4)} = \frac{\sqrt{2}}{2}\)

These computations yield the Cartesian coordinates \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). This transformation lets us plot the point on a standard grid, showing how it's critical to translate and visualize data in various forms of geometry.