Problem 15 - Calculus Early Transcendentals

Question

Express the following polar coordinates in Cartesian coordinates. \(\left(3, \frac{\pi}{4}\right)\)

Step-by-Step Solution

Answer

Answer: The Cartesian coordinates of the point with polar coordinates (3, π/4) are (3√2/2, 3√2/2).

Step 1: Identify given polar coordinates

We are given the polar coordinates \((3, \frac{\pi}{4})\), where \(r = 3\) and \(\theta = \frac{\pi}{4}\).

Step 2: Calculate x-coordinate

Using the formula \(x = r\cos\theta\), we can find the x-coordinate of the Cartesian coordinates: $$x = 3\cos\left(\frac{\pi}{4}\right)$$ The cosine of \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\), so: $$x = 3\times\frac{\sqrt{2}}{2}$$ $$x = \frac{3\sqrt{2}}{2}$$

Step 3: Calculate y-coordinate

Using the formula \(y = r\sin\theta\), we can find the y-coordinate of the Cartesian coordinates: $$y = 3\sin\left(\frac{\pi}{4}\right)$$ The sine of \(\frac{\pi}{4}\) is also \(\frac{\sqrt{2}}{2}\), so: $$y = 3\times\frac{\sqrt{2}}{2}$$ $$y = \frac{3\sqrt{2}}{2}$$

Step 4: Write the Cartesian coordinates

We have found both the x and y coordinates, so we can now write the Cartesian coordinates as: $$(x, y) = \left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$$ The polar coordinates \((3, \frac{\pi}{4})\) in Cartesian coordinates are \(\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)\).

Key Concepts

Polar Coordinates

When exploring the realm of math and geometry, polar coordinates offer a unique way to represent points on a plane. Instead of using horizontal and vertical distances to define a location, it employs the distance from a reference point called the origin, along with an angle measured from a reference direction.

A point in polar coordinates is expressed as \( (r, \theta) \), where \( r \) stands for the radial distance from the origin, and \( \theta \) is the angular component, representing the counterclockwise angle from the positive x-axis. This system is particularly useful in scenarios where symmetry relative to a point is a key feature, such as in circular or spiral patterns.

Cartesian Coordinates

In contrast to the circular flair of polar coordinates, Cartesian coordinates are all about straight lines and right angles. They form the backbone of the familiar grid system used in most graphs. A point is located by its horizontal distance (x-coordinate) and vertical distance (y-coordinate) from the origin, where the x and y axes intersect and create a four-quadrant grid.

Cartesian coordinates are denoted by \( (x, y) \). This notation is widespread across various fields including physics, engineering, and computer graphics because of its intuitive formatting and ease of use in equations.

Trigonometry in Coordinate Conversion

Trigonometry bridges the gap between polar and Cartesian coordinates. Using the trig functions sine and cosine, we can convert from one system to the other. Assuming we have a point's polar coordinates \( (r, \theta) \), the conversion formulas are:

\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
Here, \( r \) is the radial distance from the origin in the polar system, and \( \theta \) is the angle with respect to the positive x-axis. The results are the x and y coordinates for the Cartesian system, giving us a direct translation from one coordinate system to the other.

Polar Coordinate System

The polar coordinate system itself is a two-dimensional coordinate system where each point on a plane is determined by a distance and an angle. Unlike Cartesian coordinates that lay out a grid-like map, the polar system sketches out a radial lattice where each 'circle' represents points that are equidistant from the origin, and each 'line' emanating from the origin marks angles that are equal relative to the positive x-axis.

This system is superb for dealing with problems involving periodic functions, such as waves and oscillations, and is also the go-to for navigating the cyclical nature of angles and rotations in advanced mathematics and physics.