Find the exact value of each expression, if it is defined. (a) sin ^-1(1)/(2) (b) cos ^-

step 1 Okay, so we have three problems here. We're going to be finding the exact value of each expressi

Find the exact value of each expression, if it is defined. ...

Find the exact value of each expression, if it is defined.
$\begin{array}{lll}{\text { (a) } \sin ^{-1} \frac{1}{2}} & {\text { (b) } \cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)} & {\text { (c) } \tan ^{-1}(-1)}\end{array}$

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Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the basic trigonometric functions (sine, cosine, and tangent), defined in restricted domains where they are one-to-one. They allow you to find the angle corresponding to a given trigonometric value. For example, when the sine of an angle is known, the inverse sine function can be used to determine that angle.

Unit Circle and Special Angles

The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane, which is used to define trigonometric functions for all real angles. Special angles such as 30°, 45°, and 60° (or their radian measures) have well-known sine, cosine, and tangent values. Recognizing these angles is crucial for finding exact values of trigonometric expressions.

Principal Branch and Domain Restrictions

The principal branch of an inverse trigonometric function refers to the specific interval chosen so that the function is one-to-one and its inverse is well-defined. Each inverse trigonometric function has a standard range: for instance, the range of the inverse sine is typically [-?/2, ?/2], the inverse cosine is [0, ?], and the inverse tangent is (-?/2, ?/2). These restrictions ensure that each inverse function yields a unique value.

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