Inverse Trigonometric Functions

The page begins with an in-depth exploration of the inverse sine function, also known as arcsine. It explains the relationship between y = sin x and its inverse, arcsin y = x. The function's domain and range are clearly defined, emphasizing the importance of selecting the correct interval for uniqueness.

Definition: The inverse sine function, denoted as arcsin x or sin⁻¹ x, is the inverse of the sine function restricted to the domain [-π/2, π/2].

Highlight: A key property of inverse functions is highlighted: the composition of a function with its inverse yields the identity function.

An example is provided to illustrate the application of the inverse sine function in solving trigonometric equations. The solution process emphasizes the importance of considering the appropriate quadrant when dealing with negative values.

Example: For the equation sin x = -√3/2, the solution is found in the second quadrant, resulting in x = 300°.

The page then transitions to the inverse cosine function (arccos). Similar to arcsine, the domain and range are specified, and the notation arccos y = x is introduced.

Vocabulary: Arccos refers to the inverse cosine function, which is defined on the interval [0, π].

A practical problem is presented, asking students to determine the largest integer value of 'a' for which arccos(2-a) is defined. This example reinforces the understanding of the function's domain restrictions.

Lastly, the inverse tangent function (arctan) is introduced. The guide explains its domain and range, noting that it spans all real numbers for its input.

Highlight: The arctan function has a range of (-π/2, π/2), making it unique among the inverse trigonometric functions in its ability to handle any real number input.

The page concludes with a general note on the composition of inverse trigonometric functions, reinforcing the concept that a function composed with its inverse results in the identity function.