Express the value of the inverse hyperbolic function in terms of natural logarithms.
-$\operatorname { sech } ^ { - 1 } \left( \frac { 12 } { 13 } \right)$
A) $\ln \frac { 3 } { 2 }$
B) Undefined
C) $\ln \frac { 18 } { 13 }$
D) $\ln \frac { 2 } { 3 }$

Question

Quizplus · Accepted Answer

The inverse hyperbolic secant function is defined as $\operatorname { sech } ^ { - 1 } x = \ln \left( \frac { 1 } { x } + \sqrt { \frac { 1 } { x ^ { 2 } } - 1 } \right)$. Substituting $x = \frac { 12 } { 13 }$ into this equation, we get $\operatorname { sech } ^ { - 1 } \left( \frac { 12 } { 13 } \right) = \ln \left( \frac { 1 } { \frac { 12 } { 13 } } + \sqrt { \frac { 1 } { \left( \frac { 12 } { 13 } \right) ^ { 2 } } - 1 } \right) = \ln \left( \frac { 13 } { 12 } + \sqrt { \frac { 13 ^ { 2 } } { 12 ^ { 2 } } - 1 } \right) = \ln \left( \frac { 13 } { 12 } + \sqrt { \frac { 169 } { 144 } - 1 } \right) = \ln \left( \frac { 13 } { 12 } + \sqrt { \frac { 25 } { 144 } } \right) = \ln \left( \frac { 13 } { 12 } + \frac { 5 } { 12 } \right) = \ln \frac { 3 } { 2 }$.

Express the Value of the Inverse Hyperbolic Function in Terms sech⁡−1(1213)\operatorname { sech } ^ { - 1 } \left( \frac { 12 } { 13 } \right)sech−1(1312​)

Express the Value of the Inverse Hyperbolic Function in Terms $\operatorname { sech } ^ { - 1 } \left( \frac { 12 } { 13 } \right)$