θ = tan⁻¹(y/x)

Definition

The term $\theta = \tan^{-1}(y/x)$ is a fundamental concept in the context of polar coordinates. It represents the angle, measured in radians, between the positive x-axis and the line segment connecting the origin (0, 0) to a point (x, y) in the coordinate plane.

5 Must Know Facts For Your Next Test

1. The angle $\theta$ is measured counterclockwise from the positive x-axis in the polar coordinate system.
2. The value of $\theta$ can range from $-\pi$ to $\pi$ radians, or from $-180^\circ$ to $180^\circ$.
3. The polar coordinates of a point (x, y) are given by the ordered pair $(r, \theta)$, where $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(y/x)$.
4. The inverse tangent function $\tan^{-1}$ is used to find the angle $\theta$ given the coordinates (x, y) of a point.
5. Polar coordinates are often used in engineering, physics, and other fields to describe the location of objects or phenomena that have a natural radial or angular component.

Review Questions

• Explain the relationship between the Cartesian coordinates (x, y) and the polar coordinates (r, θ) of a point in the coordinate plane.
  • The polar coordinates (r, θ) of a point in the coordinate plane are directly related to its Cartesian coordinates (x, y). The radius $r$ is the distance from the origin to the point, and is calculated as $r = \sqrt{x^2 + y^2}$. The angle $\theta$ is the counterclockwise angle measured from the positive x-axis to the line segment connecting the origin to the point, and is calculated as $\theta = \tan^{-1}(y/x)$. This relationship allows for the conversion between Cartesian and polar coordinate systems, which is useful in many applications.
• Describe how the value of the angle $\theta = \tan^{-1}(y/x)$ is determined based on the signs of the x and y coordinates.
  • The value of the angle $\theta = \tan^{-1}(y/x)$ depends on the signs of the x and y coordinates. If both x and y are positive, then $\theta$ will be in the first quadrant, between 0 and $\pi/2$ radians. If x is negative and y is positive, then $\theta$ will be in the second quadrant, between $\pi/2$ and $\pi$ radians. If both x and y are negative, then $\theta$ will be in the third quadrant, between $\pi$ and $3\pi/2$ radians. If x is positive and y is negative, then $\theta$ will be in the fourth quadrant, between $3\pi/2$ and $2\pi$ radians. Understanding this relationship between the signs of the coordinates and the value of $\theta$ is crucial for properly interpreting and working with polar coordinates.
• Analyze the importance of the inverse tangent function $\tan^{-1}$ in the context of polar coordinates and explain how it is used to determine the angle $\theta$.
  • The inverse tangent function $\tan^{-1}$ is essential in the context of polar coordinates because it allows us to determine the angle $\theta$ given the Cartesian coordinates (x, y) of a point. The formula $\theta = \tan^{-1}(y/x)$ is the key to converting between Cartesian and polar coordinate systems. The inverse tangent function gives the angle whose tangent is the ratio of the y-coordinate to the x-coordinate. This is a crucial step in expressing the location of a point in polar coordinates, which are often more convenient and intuitive than Cartesian coordinates for many applications in science, engineering, and mathematics. Understanding the properties and usage of the $\tan^{-1}$ function is therefore vital for working with and interpreting polar coordinate systems.

Related terms

Polar Coordinates:

A coordinate system that specifies the location of a point in a plane using the distance from a fixed point (the origin) and the angle from a fixed direction (the positive x-axis).

Radians:

A unit of angular measurement that represents the ratio of the length of an arc to the radius of the circle. One radian is equal to the angle that subtends an arc equal in length to the radius of the circle.

Inverse Tangent (tan⁻¹): The inverse trigonometric function that gives the angle whose tangent is a given value.