Evaluating Inverse Trigonometric Functions Find the exact...

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

Key Concepts

Undefined Values in Inverse Trigonometric Functions

When working with inverse trigonometric functions, it is important to recognize that inputs falling outside the defined domain (such as for arcsin, any value beyond -1 to 1) do not yield a valid angle. These cases are considered undefined and signal that the expression cannot be evaluated within the standard framework of real numbers.

Exact Trigonometric Values

Exact trigonometric values refer to the specific angles for which the sine, cosine, or tangent exactly equal common fractional or radical numbers. Recognizing these values, often derived from special triangles or the unit circle, allows for a swift and precise evaluation of inverse trigonometric expressions.

Domain and Range of Inverse Sine Function

For the inverse sine function, the domain is limited to values between -1 and 1, inclusive, because these are the only values produced by the sine function. Its range is confined to angles between -?/2 and ?/2, ensuring that each input corresponds to a unique output, which is essential for the function to be well-defined.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle that corresponds to a given trigonometric value. They are essentially the reverse of standard trigonometric functions, and are used to solve equations where the angle is unknown but the sine, cosine, or tangent value is given.

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