Question

Write the equations that are used to express a point with Cartesian coordinates \((x, y)\) in polar coordinates.

Short answer

Answer: To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), use the following equations: 1. r = \(\sqrt{x^2 + y^2}\) 2. θ: - If x > 0 and y >= 0: θ = arctan(\(\frac{y}{x}\)) - If x < 0: θ = arctan(\(\frac{y}{x}\)) + π - If x > 0 and y < 0: θ = arctan(\(\frac{y}{x}\)) + \(2\pi\) - If x = 0 and y > 0: θ = \(\frac{\pi}{2}\) - If x = 0 and y < 0: θ = \(\frac{3\pi}{2}\)

Step-by-step solution

Finding the distance r from the origin

To find the distance r between the point (x, y) and the origin (0, 0), we use the Pythagorean theorem. The distance formula is given by: r = \(\sqrt{x^2 + y^2}\)

Finding the angle θ

To find the angle θ between the positive x-axis and the line segment connecting the origin to the point (x, y), we can use the tangent function (recall that tan(θ) = \(\frac{y}{x}\)). So, to solve for θ, we find the inverse tangent of the ratio \(\frac{y}{x}\): θ = arctan(\(\frac{y}{x}\)) However, the arctan function's range is limited between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), so we need to carefully consider the quadrant in which the point lies to determine the correct angle. 1. If x > 0 and y >= 0 (Quadrant I): θ = arctan(\(\frac{y}{x}\)) 2. If x < 0 (Quadrant II or III): θ = arctan(\(\frac{y}{x}\)) + π 3. If x > 0 and y < 0 (Quadrant IV): θ = arctan(\(\frac{y}{x}\)) + \(2\pi\) 4. If x = 0 and y > 0: θ = \(\frac{\pi}{2}\) 5. If x = 0 and y < 0: θ = \(\frac{3\pi}{2}\)

Summary of Equations

To convert a point from Cartesian coordinates (x, y) to polar coordinates (r, θ), you can use the following equations: 1. r = \(\sqrt{x^2 + y^2}\) 2. θ: - If x > 0 and y >= 0: θ = arctan(\(\frac{y}{x}\)) - If x < 0: θ = arctan(\(\frac{y}{x}\)) + π - If x > 0 and y < 0: θ = arctan(\(\frac{y}{x}\)) + \(2\pi\) - If x = 0 and y > 0: θ = \(\frac{\pi}{2}\) - If x = 0 and y < 0: θ = \(\frac{3\pi}{2}\)

Key concepts

Cartesian coordinates

Cartesian coordinates are a way to represent points on a plane using two values, usually denoted as \(x\) and \(y\). These coordinates tell you the horizontal and vertical position of a point relative to the origin (0,0), which acts like the center of the grid on your graph. \(x\) refers to how far along the horizontal axis your point is, while \(y\) signifies the distance along the vertical axis.

For example, a point (3, 4) in Cartesian coordinates means you move 3 units to the right and 4 units up from the origin. This system is especially useful in various fields such as physics, engineering, and computer graphics, as it allows us to easily plot and manipulate points and objects on a plane.

It's important to understand Cartesian coordinates because they serve as the stepping stone for converting any point to polar coordinates, which is another way to express location.

distance formula

The distance formula is used to find the distance between two points on a cartesian plane. When converting to polar coordinates, we specifically look at the distance from a point \(x, y\) to the origin \(0, 0\).

Imagine a right triangle formed by the x-axis, y-axis, and the line connecting the origin to point \(x, y\). The length of the base of the triangle is \(x\), and the height is \(y\). The distance from the origin to the point can be thought of as the hypotenuse of this triangle.

Using the Pythagorean theorem, the distance \(r\) is found using:
\[ r = \sqrt{x^2 + y^2} \]
This formula gives you the length of the hypotenuse, which is your distance in polar coordinates. It's a fundamental concept that helps bridge Cartesian and polar systems, allowing you to seamlessly transform between them.

angle calculation

In polar coordinates, the angle \(\theta\) shows the direction of the point as seen from the origin. It's usually measured from the positive x-axis.

To find this angle, we examine the relative lengths of the sides of our imaginary triangle. Using the tangent function, which is the ratio of the opposite to the adjacent side of the triangle, we get:
\[ \tan(\theta) = \frac{y}{x} \]
Solving for \(\theta\) means we need to find the angle whose tangent equals this ratio. We use the inverse tangent function, also known as arctan:
\[ \theta = \arctan\left(\frac{y}{x}\right) \]
The result gives the angle \(\theta\) but be cautious of its range. Depending on which quadrant the point sits, the angle needs adjustment to ensure it's measured correctly from the positive x-axis.

• First Quadrant: angle as is.
• Second and Third Quadrants: add \(\pi\).
• Fourth Quadrant: add \(2\pi\).

Ensuring the correct angle is key to accurately using polar coordinates.

inverse trigonometric functions

Inverse trigonometric functions are critical when transitioning between Cartesian and polar coordinates. They allow us to determine angles from known side ratios in a triangle. Specifically, the inverse tangent (arctan) function is widely used for this purpose.

The arctan function takes the ratio of the two sides of a right triangle, \(\frac{y}{x}\), and returns the angle whose tangent is that ratio. However, the range of \(\arctan\) is naturally \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This limitation requires us to adjust the \(\theta\) value when dealing with points outside the first quadrant.

In calculus and geometry, inverse trigonometric functions help in accurately determining angles, crucial for calculations in physics and engineering that involve rotations and transformations.
Understanding how these functions work ensures you can effectively translate Cartesian coordinates into polar coordinates, capturing both distance and direction.