Evaluating Inverse Trigonometric Functions Find the exact...

Evaluating Inverse Trigonometric Functions Find the exact value of each expression, if it is defined.
$$
\begin{array}{llll}{\text { (a) } \sin ^{-1}(-1)} & {\text { (b) } \sin ^{-1} \frac{\sqrt{2}}{2}} & {\text { (c) } \sin ^{-1}(-2)} & {}\end{array}
$$

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Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the inverse sine, are used to determine the angle that corresponds to a given trigonometric value. They reverse the action of the trigonometric functions, mapping a ratio back to an angle, and are defined with specific principal branches to maintain their status as functions with unique outputs.

Domain and Range of the Inverse Sine Function

The inverse sine function (sin?¹) accepts only inputs from -1 to 1, since these are the only possible outputs of the sine function for real numbers. Its output, or range, is limited to angles between -?/2 and ?/2, ensuring a unique inverse for each input within the domain.

Exact Evaluation of Trigonometric Values

Evaluating inverse trigonometric expressions exactly requires familiarity with the unit circle and standard angles. This knowledge enables one to confidently identify angles corresponding to common sine values, such as those that yield fractions like ?2/2, without approximation.

Handling Undefined Inputs

When dealing with inverse trigonometric functions, it is important to recognize when an expression is undefined. For instance, inputs outside the domain of the inverse sine function (which is [-1, 1]) do not correspond to any real angle, so the inverse function does not produce an output in such cases.

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