Arcsine

Definition

The arcsine function, denoted as $\arcsin(x)$, is the inverse of the sine function. It is used to find the angle whose sine is equal to a given value.

5 Must Know Facts For Your Next Test

1. The arcsine function is used to find the angle $\theta$ such that $\sin(\theta) = x$, where $-1 \leq x \leq 1$.
2. The domain of the arcsine function is $[-1, 1]$, and the range is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.
3. The arcsine function is often denoted as $\arcsin(x)$ or $\sin^{-1}(x)$, both of which represent the same inverse trigonometric function.
4. Integrals involving the arcsine function can be evaluated using the substitution $u = \sin(x)$, which leads to $du = \cos(x) dx$.
5. The derivative of the arcsine function is $\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}$.

Review Questions

• Explain the relationship between the sine function and the arcsine function.
  • The sine function and the arcsine function are inverse functions. The sine function takes an angle as input and returns the ratio of the opposite side to the hypotenuse of a right triangle. The arcsine function, on the other hand, takes a value between -1 and 1 as input and returns the angle whose sine is that value. In other words, if $y = \sin(x)$, then $x = \arcsin(y)$. This inverse relationship allows us to find the angle given the value of the sine function, which is particularly useful in applications involving trigonometry.
• Describe how the arcsine function is used in the context of integrals resulting in inverse trigonometric functions.
  • In the context of integrals resulting in inverse trigonometric functions, the arcsine function is used as a substitution to simplify the integration process. When an integral involves an expression containing the sine function, we can substitute $u = \sin(x)$, which leads to $du = \cos(x) dx$. This substitution transforms the integral into one involving the arcsine function, which can then be evaluated using the properties of inverse trigonometric functions. The arcsine function allows us to express the integral in a more manageable form and ultimately find the antiderivative of the original expression.
• Analyze the relationship between the domain and range of the arcsine function and its applications in calculus.
  • The domain of the arcsine function is $[-1, 1]$, which means that the input values must be within this range for the function to be defined. The range of the arcsine function is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, which represents the set of all possible output values. This relationship between the domain and range is crucial in the context of integrals resulting in inverse trigonometric functions. The restricted domain of the arcsine function ensures that the substitution $u = \sin(x)$ is valid, and the range of the arcsine function allows us to express the integral in terms of the inverse trigonometric function. Understanding the properties of the arcsine function, including its domain and range, is essential for successfully applying it in the integration of expressions involving the sine function.

Related terms

Inverse Trigonometric Functions:

Inverse trigonometric functions are the inverse of the basic trigonometric functions, such as sine, cosine, and tangent. They allow you to find the angle given the value of the trigonometric function.

Trigonometric Identities:

Trigonometric identities are equations that relate the trigonometric functions to one another. They are essential for simplifying and manipulating expressions involving trigonometric functions.

Integration by Substitution: Integration by substitution, also known as the method of u-substitution, is a technique used to evaluate integrals by replacing the original variable with a new variable that makes the integral easier to solve.