Question

simplify sinh(ln(t)) by rewriting it into a rational expression.
sinh(ln(t)) =

Explanation:

Step1: Recall the definition of hyperbolic sine

The definition of \(\sinh(x)=\frac{e^{x}-e^{-x}}{2}\). So, \(\sinh(\ln(t))=\frac{e^{\ln(t)}-e^{-\ln(t)}}{2}\).

Step2: Simplify the exponential - logarithmic expressions

We know that \(e^{\ln(t)} = t\) and \(e^{-\ln(t)}=e^{\ln(t^{-1})}=t^{-1}=\frac{1}{t}\) (using the properties \(a\ln(b)=\ln(b^{a})\) and \(e^{\ln(u)} = u\) for \(u>0\)).
Substitute these into the expression: \(\sinh(\ln(t))=\frac{t-\frac{1}{t}}{2}\).

Step3: Simplify the rational expression

\(\frac{t-\frac{1}{t}}{2}=\frac{\frac{t^{2}-1}{t}}{2}=\frac{t^{2}-1}{2t}\) (by getting a common denominator in the numerator and then dividing by 2).

Answer:

\(\frac{t^{2}-1}{2t}\)