Arctan

Definition

Arctan, also known as the inverse tangent function, is a trigonometric function that calculates the angle whose tangent is a given value. It is the inverse operation of the tangent function, allowing one to determine the angle given the tangent ratio.

5 Must Know Facts For Your Next Test

1. The arctan function is used to find the angle given the tangent ratio, which is useful in trigonometric substitution problems.
2. The domain of the arctan function is the set of all real numbers, and the range is the interval $(-\pi/2, \pi/2)$.
3. The arctan function is often denoted as $\tan^{-1}(x)$, where $x$ is the tangent ratio.
4. Trigonometric substitution using the arctan function is commonly employed in integration problems involving square roots of quadratic expressions.
5. The derivative of the arctan function is $\frac{1}{1 + x^2}$, which is useful in differentiation problems involving the arctan function.

Review Questions

• Explain how the arctan function is used in the context of trigonometric substitution.
  • In trigonometric substitution, the arctan function is used to replace a variable in an integral involving a square root of a quadratic expression. The substitution $x = \tan(\theta)$ allows the integral to be rewritten in terms of the variable $\theta$, where $\arctan(x) = \theta$. This transformation simplifies the integral and often leads to easier integration techniques, such as the use of trigonometric identities or integration by parts.
• Describe the properties of the arctan function, including its domain, range, and derivative.
  • The arctan function, also denoted as $\tan^{-1}(x)$, has a domain of all real numbers and a range of $(-\pi/2, \pi/2)$. This means that the arctan function can accept any real number as input and will output an angle within the interval $(-\pi/2, \pi/2)$ radians. The derivative of the arctan function is $\frac{1}{1 + x^2}$, which is useful in differentiation problems involving the arctan function.
• Analyze how the properties of the arctan function, such as its domain and range, impact the use of trigonometric substitution in integration problems.
  • The properties of the arctan function, particularly its domain and range, are crucial in the context of trigonometric substitution. The fact that the arctan function has a domain of all real numbers allows for a wide range of substitutions to be made in integration problems involving square roots of quadratic expressions. Additionally, the restricted range of $(-\pi/2, \pi/2)$ ensures that the substituted variable $\theta$ remains within a manageable interval, simplifying the integration process and leading to more tractable solutions. These properties of the arctan function make it a valuable tool in the application of trigonometric substitution in calculus.

Related terms

Tangent: The tangent function is a trigonometric function that calculates the ratio of the opposite side to the adjacent side of a right triangle.

Inverse Function: An inverse function is a function that undoes the operation of another function, returning the original input when the output of the first function is provided.

Trigonometric Substitution: Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions by replacing the variable with a trigonometric function.