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📏honors pre-calculus review

key term - θ = tan⁻¹(y/x)

Citation:

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.