Question

69-74: Find the exact value of each expression.

69.

(a) \({\cos ^{ - 1}}\left( { - 1} \right)\)

(b) \({\sin ^{ - 1}}\left( {0.5} \right)\)

Short answer

(a) The exact value of \({\cos ^{ - 1}}\left( { - 1} \right)\) is \(\pi \).

(b) The exact value of \({\sin ^{ - 1}}\left( {0.5} \right)\) is \(\frac{\pi }{6}\).

Step-by-step solution

Inverse sine function and inverse cosine function

If we restrict the domain to the interval \(\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)\), therefore, the function is one-to-one. The inverse function of the restricted sine function\(f\) is denoted by \({\sin ^{ - 1}}\) or \(\arcsin \).

\({\sin ^{ - 1}}x = y \Leftrightarrow \sin y = x\,\,\,\,{\mathop{\rm and}\nolimits} \,\,\,\, - \frac{\pi }{2} \le y \le \frac{\pi }{2}\).

The restricted cosine function \(y = \cos x,0 \le x \le \pi \) is a one-to-one function, and its inverse function is denoted by \({\cos ^{ - 1}}\) or \(\arccos \).

\({\cos ^{ - 1}}x = y \Leftrightarrow \cos y = x\,\,\,\,{\mathop{\rm and}\nolimits} \,\,\,\,0 \le y \le \pi \).

Determine the exact value of the expression

a)

Determine the exact value of the expression as shown below:

\({\cos ^{ - 1}}\left( { - 1} \right) = \pi \)

Since \(\cos \pi = - 1\) and \(\pi \) is in the interval \(\left( {0,\pi } \right)\) (the range of \({\cos ^{ - 1}}\)).

Thus, the exact value of the expression is \(\pi \).

b)

Determine the exact value of the expression as shown below:

\({\sin ^{ - 1}}\left( {0.5} \right) = \frac{\pi }{6}\)

Since \(\sin \frac{\pi }{6} = 0.5\) and \(\frac{\pi }{6}\) is in the interval \(\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)\) (the range of \({\sin ^{ - 1}}\)).

Thus, the exact value of the expression is \(\frac{\pi }{6}\).