Question

Find the exact value of each expression.

73. (a) \({\cot ^{ - 1}}\left( { - \sqrt 3 } \right)\) (b) \({\sec ^{ - 1}}2\)

Short answer

(a) The exact value of \({\cot ^{ - 1}}\left( { - \sqrt 3 } \right)\) is \(\frac{{5\pi }}{6}\).

(b) The exact value of \({\sec ^{ - 1}}2\) is \(\frac{\pi }{3}\).

Step-by-step solution

Inverse trigonometric function

The remaining inverse trigonometricfunctions are shown below:

\(\begin{aligned}y &= {\csc ^{ - 1}}x\left( {\left| x \right| \ge 1} \right) \Leftrightarrow \csc y = x\,\,\,\,{\mathop{\rm and}\nolimits} \,\,\,y \in \left( {0,\frac{\pi }{2}} \right) \cup \left( {\pi ,\frac{{3\pi }}{2}} \right)\\y &= {\sec ^{ - 1}}x\left( {\left| x \right| \ge 1} \right) \Leftrightarrow \sec y = x\,\,\,{\mathop{\rm and}\nolimits} \,\,\,y \in \left( {0,\frac{\pi }{2}} \right) \cup \left( {\pi ,\frac{{3\pi }}{2}} \right)\\y &= {\cot ^{ - 1}}x\left( {x \in \mathbb{R}} \right) \Leftrightarrow \cot y = x\,\,\,\,\,{\mathop{\rm and}\nolimits} \,\,\,y \in \left( {0,\pi } \right)\end{aligned}\)

Determine the exact value of the expression 

a)

Determine the exact value of the expression as shown below:

\({\cot ^{ - 1}}\left( { - \sqrt 3 } \right) = \frac{{5\pi }}{6}\)

Since \(\cot \left( {\frac{{5\pi }}{6}} \right) = - \sqrt 3 \) and \(\frac{{5\pi }}{6}\) is in the interval \(\left( {0,\pi } \right)\). (The range of \({\cot ^{ - 1}}\)).

Thus, the exact value of \({\cot ^{ - 1}}\left( { - \sqrt 3 } \right)\) is \(\frac{{5\pi }}{6}\).

b)

Determine the exact value of the expression as shown below:

\({\sec ^{ - 1}}2 = \frac{\pi }{3}\)

Since \(\sec \frac{\pi }{3} = 2\) and \(\frac{\pi }{3}\) is in the interval\(\left( {0,\frac{\pi }{2}} \right) \cup \left( {\pi ,\frac{{3\pi }}{2}} \right)\) (the range of \({\sec ^{ - 1}}\)).

Thus, the exact value of \({\sec ^{ - 1}}2\) is \(\frac{\pi }{3}\).