Arctangent function

Definition

The arctangent function, often denoted as \(\tan^{-1}(x)\) or \(\text{atan}(x)\), is the inverse of the tangent function. It returns the angle whose tangent is the given number, mapping from the real numbers to the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\). This function is essential in trigonometry as it helps determine angles from tangent values, connecting it to the broader context of trigonometric functions and their applications.

5 Must Know Facts For Your Next Test

1. The range of the arctangent function is limited to \((-\frac{\pi}{2}, \frac{\pi}{2})\), meaning it outputs angles only in this interval.
2. The arctangent function has a vertical asymptote at both ends of its range, which means as \(x\) approaches positive or negative infinity, the output approaches \(\frac{\pi}{2}\) or \(-\frac{\pi}{2}\).
3. The derivative of the arctangent function is given by \(f'(x) = \frac{1}{1+x^2}\), which means it is always positive and thus monotonically increasing.
4. The arctangent can also be expressed in terms of logarithms: \(\tan^{-1}(x) = \frac{1}{2i} \left(\ln(1 + ix) - \ln(1 - ix)\right)\).
5. Using the arctangent function, you can easily find angles in right triangles when you know the lengths of the opposite and adjacent sides.

Review Questions

• How does the arctangent function relate to the tangent function in terms of inverse relationships?
  • The arctangent function serves as the inverse of the tangent function. This means that if you have an angle \(y = \tan^{-1}(x)\), applying tangent on both sides gives you \(x = \tan(y)\). This relationship highlights how arctangent can be used to recover angles from known tangent values.
• Discuss how the properties of the arctangent function influence its graph, particularly regarding asymptotes and monotonicity.
  • The graph of the arctangent function features horizontal asymptotes at \(y = \frac{\pi}{2}\) and \(y = -\frac{\pi}{2}\), which means that as you move towards positive or negative infinity on the x-axis, the y-values approach these limits but never actually reach them. Additionally, since its derivative is always positive, this indicates that the function is monotonically increasing throughout its entire domain.
• Evaluate how understanding the arctangent function contributes to solving real-world problems involving angles and distances.
  • Understanding the arctangent function is crucial in fields such as physics, engineering, and navigation where determining angles from ratios of sides is essential. For instance, if you know the height and distance from an object, you can use arctangent to calculate its angle of elevation. This application illustrates not only its theoretical importance but also its practical utility in various problem-solving scenarios.

Related terms

Tangent function: A trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle, often represented as \(\tan(x) = \frac{\text{opposite}}{\text{adjacent}}\).

Inverse functions: Functions that reverse the effect of the original function, such that if \(f(x) = y\), then the inverse function \(f^{-1}(y) = x\).

Unit circle: A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions and their relationships to angles.