5.1: Hyperbolic Functions (Lecture Notes for Augmented Lectures)
- Page ID
- 145454
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Let \( \frac{t}{2} \) be the area bounded by the right branch of the unit hyperbola \( x^2 - y^2 = 1 \) whose initial point is \( \left( 1,0 \right) \) and terminal point is \( \left( x,y \right) \). Then the hyperbolic functions are defined (and derived) as follows:
Hyperbolic cosine
\(x = \cosh{(t)} = \dfrac{e^t+e^{−t}}{2}\)
Hyperbolic sine
\(y = \sinh{(t)} = \dfrac{e^t−e^{−t}}{2}\)
Hyperbolic tangent
\(\tanh{(t)} = \dfrac{\sinh{(t)}}{\cosh{(t)}} = \dfrac{e^t−e^{−t}}{e^t+e^{−t}}\)
Hyperbolic cosecant
\(\operatorname{csch}{(t)} = \dfrac{1}{\sinh{(t)}} = \dfrac{2}{e^t−e^{−t}}\)
Hyperbolic secant
\(\operatorname{sech}{(t)} = \dfrac{1}{\cosh{(t)}} = \dfrac{2}{e^t+e^{−t}}\)
Hyperbolic cotangent
\(\coth{(t)} = \dfrac{\cosh{(t)}}{\sinh{(t)}} = \dfrac{e^t+e^{−t}}{e^t−e^{−t}}\)
Hyperbolic Identities
If \(\tanh{(t)} = \frac{12}{13}\), find the values of the remaining five hyperbolic functions at \( t \).
Inverse Hyperbolic Functions
\[\begin{align*} &\sinh^{−1}x =\operatorname{arcsinh}x=\ln \left(x+\sqrt{x^2+1}\right) & & \cosh^{−1}x =\operatorname{arccosh}x=\ln \left(x+\sqrt{x^2−1}\right)\\[4pt]
&\tanh^{−1}x=\operatorname{arctanh}x=\dfrac{1}{2}\ln \left(\dfrac{1+x}{1−x}\right) & & \coth^{−1}x =\operatorname{arccot}x=\frac{1}{2}\ln \left(\dfrac{x+1}{x−1}\right)\\[4pt]
&\operatorname{sech}^{−1}x=\operatorname{arcsech}x=\ln \left(\dfrac{1+\sqrt{1−x^2}}{x}\right) & & \operatorname{csch}^{−1}x=\operatorname{arccsch}x=\ln \left(\dfrac{1}{x}+\dfrac{\sqrt{1+x^2}}{|x|}\right) \end{align*}\]