Question

For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3.

(a) \({\bf{csc}}{{\bf{h}}^{ - {\bf{1}}}}\) (b) \({\bf{sec}}{{\bf{h}}^{ - {\bf{1}}}}\) (c) \({\bf{cot}}{{\bf{h}}^{ - {\bf{1}}}}\)

Short answer

(a) (i) \({\mathop{\rm csch}\nolimits} y = x\)

(iii) \({{\mathop{\rm csch}\nolimits} ^{ - 1}}x = \ln \left( {\frac{1}{x} + \frac{{\sqrt {{x^2} + 1} }}{{\left| x \right|}}} \right)\)

(b) (i) \({\mathop{\rm sech}\nolimits} y = x\)

(iii) \({{\mathop{\rm sech}\nolimits} ^{ - 1}}x = \ln \left( {\frac{1}{x} + \frac{{\sqrt {1 - {x^2}} }}{x}} \right)\)

(c) (i) \(x = \coth y\)

(iii) \({\coth ^{ - 1}}x = \frac{1}{2}\ln \left( {\frac{{x + 1}}{{x - 1}}} \right)\)

Step-by-step solution

Step 1:Find answer for part (a)

(i) Let \(y = {{\mathop{\rm csch}\nolimits} ^{ - 1}}x\), then\({\mathop{\rm csch}\nolimits} y = x\).

(ii) Use the following steps to plot the graph of given functions:

1. In the graphing calculator select “STAT PLOT” and enter the equation \({{\mathop{\rm csch}\nolimits} ^{ - 1}}x\).
2. Enter the graph button in the graphing calculator.

The figure below represents the graph of \({{\mathop{\rm csch}\nolimits} ^{ - 1}}x\).

(iii) Let \(y = {{\mathop{\rm csch}\nolimits} ^{ - 1}}x\), then

\(\begin{aligned}x &= {\mathop{\rm csch}\nolimits} y\\x &= \frac{2}{{{e^y} - {e^{ - y}}}}\\x{e^y} - x{e^{ - y}} &= 2\\x{\left( {{e^y}} \right)^2} - 2{e^y} - x &= 0\\{e^y} &= \frac{{1 \pm \sqrt {{x^2} + 1} }}{x}\end{aligned}\)

For \(x > 0\),

\({e^y} = \frac{{1 + \sqrt {{x^2} + 1} }}{x}\)

For \(x < 0\),

\({e^y} = \frac{{1 - \sqrt {{x^2} + 1} }}{x}\)

The value of \({{\mathop{\rm csch}\nolimits} ^{ - 1}}x\) is\({{\mathop{\rm csch}\nolimits} ^{ - 1}}x = \ln \left( {\frac{1}{x} + \frac{{\sqrt {{x^2} + 1} }}{{\left| x \right|}}} \right)\).

Find answer for part (b)

(i) Let \(y = {{\mathop{\rm sech}\nolimits} ^{ - 1}}x\), then\({\mathop{\rm sech}\nolimits} y = x\).

(ii) Use the following steps to plot the graph of given functions:

1. In the graphing calculator select “STAT PLOT” and enter the equation \({{\mathop{\rm sech}\nolimits} ^{ - 1}}x\).
2. Enter the graph button in the graphing calculator.

(iii) Let \(y = {{\mathop{\rm sech}\nolimits} ^{ - 1}}x\), then

\(\begin{aligned}x&= {\mathop{\rm sech}\nolimits} y\\x &= \frac{2}{{{e^y} + {e^{ - y}}}}\\x{e^y} + x{e^{ - y}} &= 2\\x{\left( {{e^y}} \right)^2} - 2{e^y} + x &= 0\\{e^y} &= \frac{{1 \pm \sqrt {1 - {x^2}} }}{x}\end{aligned}\)

If,

\(\begin{aligned}\frac{{1 - \sqrt {1 - {x^2}} }}{x} > 1\\1 - \sqrt {1 - {x^2}} > x\\1 - 2x + {x^2} > 1 - {x^2}\\{x^2} > x\\x > 1\end{aligned}\)

But value of \({\mathop{\rm sech}\nolimits} y \le 1\).

Thus,

\(\begin{aligned}{e^y} = \frac{{1 + \sqrt {1 - {x^2}} }}{x}\\{{\mathop{\rm sech}\nolimits} ^{ - 1}}x = \ln \left( {\frac{1}{x} + \frac{{\sqrt {1 - {x^2}} }}{x}} \right)\end{aligned}\)

Find the answer for part (c)

(i) Let \(y = {\coth ^{ - 1}}x\), then\(x = \coth y\).

(ii) Use the following steps to plot the graph of given functions:

1. In the graphing calculator select “STAT PLOT” and enter the equation \({\coth ^{ - 1}}x\).
2. Enter the graph button in the graphing calculator.

The figure below represents the graph of \({\coth ^{ - 1}}x\).

(iii) Let \(y = {\coth ^{ - 1}}x\), then

\(\begin{aligned}x &= \coth y\\x &= \frac{{{e^y} + {e^{ - y}}}}{{{e^y} - {e^{ - y}}}}\\x{e^y} - x{e^{ - y}} &= {e^y} + {e^{ - y}}\\\left( {x - 1} \right){e^y} &= \left( {x + 1} \right){e^{ - y}}\\{e^{2y}} &= \frac{{x + 1}}{{x - 1}}\\y &= \frac{1}{2}\ln \left( {\frac{{x + 1}}{{x - 1}}} \right)\\{\coth ^{ - 1}}x &= \frac{1}{2}\ln \left( {\frac{{x + 1}}{{x - 1}}} \right)\end{aligned}\)

Thus, \({\coth ^{ - 1}}x = \frac{1}{2}\ln \left( {\frac{{x + 1}}{{x - 1}}} \right)\).