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📈college algebra review

Inverse Sine Identity

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The inverse sine identity, also known as the arcsine identity, is a fundamental relationship in trigonometry that connects the inverse sine function to the original sine function. It allows for the simplification and evaluation of expressions involving the inverse sine function.

5 Must Know Facts For Your Next Test

  1. The inverse sine identity states that $\sin^{-1}(\sin(x)) = x$, where $x$ is an angle measured in radians and $\sin^{-1}$ represents the inverse sine function.
  2. The inverse sine identity is valid for any angle $x$ in the domain of the sine function, which is the interval $[-\pi/2, \pi/2]$.
  3. The inverse sine identity is useful for simplifying expressions involving the inverse sine function, as it allows for the elimination of the inverse sine operation.
  4. The inverse sine identity is a special case of the more general inverse trigonometric identity, which states that $\theta = \text{arcsin}(\sin(\theta))$.
  5. Understanding the inverse sine identity is crucial for solving equations and inequalities involving the inverse sine function, as well as for evaluating inverse sine expressions.

Review Questions

  • Explain the significance of the inverse sine identity in the context of inverse trigonometric functions.
    • The inverse sine identity, $\sin^{-1}(\sin(x)) = x$, is a fundamental relationship that connects the inverse sine function to the original sine function. This identity is important because it allows for the simplification of expressions involving the inverse sine function by eliminating the inverse operation. Understanding this identity is crucial for solving equations and inequalities that include the inverse sine function, as well as for evaluating inverse sine expressions. The identity is valid for any angle $x$ within the domain of the sine function, which is the interval $[-\pi/2, \pi/2]$.
  • Describe how the inverse sine identity relates to the concept of radian measure.
    • The inverse sine identity, $\sin^{-1}(\sin(x)) = x$, is closely connected to the concept of radian measure. Radian measure is a way of expressing angles in terms of the ratio of the arc length to the radius of a circle, where one radian is the angle that subtends an arc equal in length to the radius of the circle. The inverse sine function, $\sin^{-1}$, is used to determine the angle (in radians) given the value of the sine function. The inverse sine identity demonstrates that the angle $x$ (in radians) is equal to its own sine value, which highlights the relationship between the inverse sine function and the radian measure of an angle.
  • Analyze how the inverse sine identity can be used to simplify expressions involving the inverse sine function.
    • The inverse sine identity, $\sin^{-1}(\sin(x)) = x$, can be used to simplify expressions involving the inverse sine function by eliminating the inverse operation. For example, if an expression contains the term $\sin^{-1}(\sin(y))$, the inverse sine identity can be applied to rewrite this as simply $y$. This simplification is useful when solving equations or inequalities that include the inverse sine function, as it allows for the elimination of the inverse operation and the direct evaluation of the expression. Additionally, the inverse sine identity can be used to evaluate inverse sine expressions by substituting the appropriate angle value for $x$, as long as the angle is within the domain of the sine function.