Arctangent Function

Definition

The arctangent function, denoted as $\tan^{-1}(x)$ or $\arctan(x)$, is an inverse trigonometric function that calculates the angle whose tangent is the given value. It is used to determine the angle from the known ratio of the opposite and adjacent sides of a right triangle.

5 Must Know Facts For Your Next Test

1. The arctangent function is used to find the angle when the ratio of the opposite and adjacent sides of a right triangle is known.
2. The arctangent function returns an angle in the range of $-\pi/2$ to $\pi/2$ radians, or $-90^\circ$ to $90^\circ$ degrees.
3. The arctangent function is often used in vector addition and subtraction problems to determine the direction of a resultant vector.
4. The arctangent function can be used to find the angle of inclination or elevation of an object relative to the horizontal.
5. The arctangent function is a continuous function, meaning it has no discontinuities or jumps in its graph.

Review Questions

• Explain how the arctangent function is used in the context of vector addition and subtraction.
  • In the topic of vector addition and subtraction, the arctangent function is used to determine the direction of the resultant vector. By taking the arctangent of the ratio of the $y$-component to the $x$-component of the vector, you can calculate the angle between the resultant vector and the positive $x$-axis. This allows you to fully characterize the magnitude and direction of the vector, which is essential for analyzing the sum or difference of multiple vectors.
• Describe how the properties of the arctangent function, such as its range and continuity, are important in the context of vector addition and subtraction.
  • The range of the arctangent function, from $-\pi/2$ to $\pi/2$ radians, is crucial in vector addition and subtraction problems because it allows you to determine the angle of the resultant vector within the full 360-degree circle. The continuity of the arctangent function ensures that small changes in the vector components result in small changes in the calculated angle, which is important for accurately analyzing the direction of the vector. These properties of the arctangent function make it a valuable tool for solving vector problems involving analytical methods.
• Evaluate how the arctangent function could be used to determine the angle of inclination or elevation of an object relative to the horizontal, and explain the significance of this application in the context of vector addition and subtraction.
  • The arctangent function can be used to find the angle of inclination or elevation of an object relative to the horizontal, which is important in the context of vector addition and subtraction. By taking the arctangent of the ratio of the vertical component to the horizontal component of the object's position or motion, you can determine the angle between the object's path and the horizontal. This information is crucial for analyzing the direction and magnitude of vectors involved in the object's movement, such as velocity or acceleration vectors. Understanding the angle of inclination or elevation allows you to properly decompose and combine vectors, which is essential for solving problems involving vector addition and subtraction using analytical methods.

Related terms

Inverse Trigonometric Functions: Inverse trigonometric functions, such as the arctangent, are used to find the angle given the value of a trigonometric ratio, unlike the regular trigonometric functions that find the ratio given the angle.

Right Triangle: A right triangle is a triangle in which one of the angles is a 90-degree angle. The arctangent function is closely related to the properties of right triangles, as it involves the ratio of the opposite and adjacent sides.

Tangent Ratio: The tangent ratio is the ratio of the opposite side to the adjacent side of a right triangle. The arctangent function is the inverse of the tangent function, allowing you to find the angle given the tangent ratio.