Inverse Trigonometric Function

Definition

An inverse trigonometric function is a function that reverses the operation of a trigonometric function, allowing the original angle to be recovered from the given trigonometric ratio. These functions are used to solve trigonometric equations by finding the angle that corresponds to a given trigonometric value.

5 Must Know Facts For Your Next Test

1. Inverse trigonometric functions are denoted with the prefix 'arc' (e.g., arcsin, arccos, arctan).
2. The domain of an inverse trigonometric function is limited to the range of the corresponding trigonometric function.
3. Inverse trigonometric functions are used to find the angle given the value of a trigonometric ratio.
4. The graphs of inverse trigonometric functions are the reflections of the original trigonometric functions across the line $y = x$.
5. Inverse trigonometric functions are essential for solving trigonometric equations, as they allow the isolation of the unknown angle.

Review Questions

• Explain how inverse trigonometric functions are used to solve trigonometric equations.
  • Inverse trigonometric functions are used to solve trigonometric equations by isolating the unknown angle. For example, if the equation is $ extbackslash sin(x) = 0.5$, we can apply the inverse sine function (arcsin) to both sides to find the angle: $x = extbackslash arcsin(0.5)$. This allows us to determine the specific angle that satisfies the given trigonometric relationship.
• Describe the relationship between the domain and range of a trigonometric function and its inverse.
  • The domain of an inverse trigonometric function is the range of the corresponding trigonometric function, and the range of the inverse trigonometric function is the domain of the original trigonometric function. This relationship is a result of the inverse function reversing the operation of the original function. For example, the domain of $\textbackslash sin^{-1}(x)$ is $[-1, 1]$, which is the range of the $\textbackslash sin(x)$ function.
• Analyze the graphical properties of inverse trigonometric functions and how they relate to the original trigonometric functions.
  • The graphs of inverse trigonometric functions are the reflections of the original trigonometric function graphs across the line $y = x$. This means that the $x$-axis and $y$-axis of the inverse function graph are switched compared to the original function. Additionally, the domain and range of the inverse function are interchanged with the range and domain of the original function, respectively. These graphical properties illustrate how the inverse function reverses the operation of the original trigonometric function.

Related terms

Trigonometric Functions: The basic trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) that relate the angles of a right triangle to the lengths of its sides.

Domain and Range:

The set of input values (domain) and output values (range) for which a function is defined.

Inverse Function:

A function that reverses the operation of another function, so that if $f(x) = y$, then $f^{-1}(y) = x$.