Arccos

Definition

The inverse cosine function, denoted as arccos or cos^-1, is a trigonometric function that gives the angle whose cosine is equal to a given real number. It is the inverse of the cosine function, allowing one to find the angle given the cosine value.

5 Must Know Facts For Your Next Test

1. The arccos function is used to find the angle whose cosine is equal to a given real number between -1 and 1.
2. The arccos function is the inverse of the cosine function, so $\arccos(\cos(x)) = x$ for all $x$ in the domain of the cosine function.
3. The derivative of the arccos function is $\frac{-1}{\sqrt{1-x^2}}$, which is an important result in the study of derivatives of inverse functions.
4. The domain of the arccos function is the interval $[-1, 1]$, as the cosine function has a range of $[-1, 1]$.
5. The range of the arccos function is the interval $[0, \pi]$, as the angles whose cosine is between -1 and 1 are between 0 and $\pi$ radians.

Review Questions

• Explain how the arccos function is the inverse of the cosine function and describe its key properties.
  • The arccos function is the inverse of the cosine function, meaning that it undoes the operation of the cosine function. If $y = \cos(x)$, then $x = \arccos(y)$. The arccos function has a domain of $[-1, 1]$, as the cosine function has a range of $[-1, 1]$. The range of the arccos function is $[0, \pi]$, as the angles whose cosine is between -1 and 1 are between 0 and $\pi$ radians. This inverse relationship between the cosine and arccos functions is a fundamental concept in trigonometry and calculus.
• Derive the formula for the derivative of the arccos function and explain its significance in the context of derivatives of inverse functions.
  • The derivative of the arccos function is $\frac{-1}{\sqrt{1-x^2}}$. This formula is derived using the chain rule and the fact that the derivative of the cosine function is $-\sin(x)$. The derivative of the arccos function is an important result in the study of derivatives of inverse functions, as it allows one to find the rate of change of the arccos function at a given point. This concept is particularly relevant in the context of Section 3.7, which discusses derivatives of inverse functions.
• Describe how the properties of the arccos function, such as its domain and range, are related to the properties of the cosine function and the behavior of trigonometric functions in general.
  • The arccos function is intimately connected to the properties of the cosine function and the behavior of trigonometric functions. The domain of the arccos function, $[-1, 1]$, corresponds to the range of the cosine function, as the arccos function is the inverse of the cosine function. Similarly, the range of the arccos function, $[0, \pi]$, corresponds to the angles whose cosine is between -1 and 1. These relationships between the arccos function and the cosine function, as well as the underlying properties of trigonometric functions, are fundamental to understanding the behavior and applications of inverse trigonometric functions in calculus and beyond.

Related terms

Inverse Function: An inverse function is a function that undoes the operation of another function. The arccos function is the inverse of the cosine function, allowing one to find the angle given the cosine value.

Derivative: The derivative of a function represents the rate of change of the function at a given point. The derivative of the inverse cosine function, arccos, is an important concept in calculus.

Domain and Range: The domain of the arccos function is the interval [-1, 1], as the cosine function has a range of [-1, 1]. The range of the arccos function is the interval [0, π].