Question

Find the exact value of each expression.

74.

(a) \(\arcsin \left( {\sin \frac{{5\pi }}{4}} \right)\)

(b) \(\cos \left( {2{{\sin }^{ - 1}}\left( {\frac{5}{{13}}} \right)} \right)\)

Short answer

a) The exact value of the expression is \( - \frac{\pi }{4}\).

b) The exact value of the expression is \(\frac{{119}}{{169}}\).

Step-by-step solution

Inverse trigonometric function

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 restrictedcosine 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)

The exact value of the expression is shown below:

\(\begin{aligned}{c}\arcsin \left( {\sin \frac{{5\pi }}{4}} \right) = \arcsin \left( {\frac{{ - 1}}{{\sqrt 2 }}} \right)\\ = - \frac{\pi }{4}\end{aligned}\)

Since \(\sin \left( { - \frac{\pi }{4}} \right) = - \frac{1}{{\sqrt 2 }}\) and \( - \frac{\pi }{4}\) is in the interval \(\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)\). (the range of \({\cot ^{ - 1}}\)).

Thus, the exact value of the expression is \( - \frac{\pi }{4}\).

b)

Consider \(\theta = {\sin ^{ - 1}}\left( {\frac{5}{{13}}} \right)\).

Draw the right triangle with angle \(\theta \) as shown below:

The exact value of the expression is shown below:

\(\cos \left( {2{{\sin }^{ - 1}}\left( {\frac{5}{{12}}} \right)} \right) = \cos 2\theta \,\,\,\,\,\,\,{\rm{ }}\left\{ {\cos 2\theta = {{\cos }^2}\theta - {{\sin }^2}\theta } \right\}\)

\( = {\cos ^2}\theta - {\sin ^2}\theta \)

\( = {\left( {\frac{{12}}{{13}}} \right)^2} - {\left( {\frac{5}{{13}}} \right)^2}\)

\( = \frac{{144}}{{169}} - \frac{{25}}{{169}}\)

\( = \frac{{119}}{{169}}\)

Thus, the exact value of the expression is \(\frac{{119}}{{169}}\).