the inverse hyperbolic secant function can be expressed as $\text{sech}^{-1}x=lnleft(\frac{1 + sqrt{1…

the inverse hyperbolic secant function can be expressed as $\text{sech}^{-1}x=lnleft(\frac{1 + sqrt{1 - x^{2}}}{x}\right)$, $0 < xleq1$. use this formula to evaluate the following expression. round your answer to two decimal places. if the value does not exist, write dne for your answer.\n$\text{sech}^{-1}(0.4)$

the inverse hyperbolic secant function can be expressed as $\text{sech}^{-1}x=lnleft(\frac{1 + sqrt{1 - x^{2}}}{x}\right)$, $0 < xleq1$. use this formula to evaluate the following expression. round your answer to two decimal places. if the value does not exist, write dne for your answer.\n$\text{sech}^{-1}(0.4)$

Answer

Explanation:

Step1: Substitute $x = 0.4$ into formula

We substitute $x = 0.4$ into $\text{sech}^{-1}x=\ln\left(\frac{1 + \sqrt{1 - x^{2}}}{x}\right)$. So we get $\text{sech}^{-1}(0.4)=\ln\left(\frac{1+\sqrt{1-(0.4)^{2}}}{0.4}\right)$.

Step2: Calculate value inside square - root

First, calculate $1-(0.4)^{2}=1 - 0.16 = 0.84$. Then $\sqrt{1-(0.4)^{2}}=\sqrt{0.84}\approx0.9165$.

Step3: Calculate value inside natural - logarithm

Next, calculate $1+\sqrt{1-(0.4)^{2}}=1 + 0.9165=1.9165$. Then $\frac{1+\sqrt{1-(0.4)^{2}}}{0.4}=\frac{1.9165}{0.4}=4.79125$.

Step4: Calculate natural - logarithm

Finally, $\text{sech}^{-1}(0.4)=\ln(4.79125)\approx1.57$.

Answer:

$1.57$