`sinh(3)` Evaluate the function. If the value is not a rational number, round your answer to three decimal places.

Tuesday, July 19, 2011

$\displaystyle{\sinh{{\left({3}\right)}}}$

Take note that hyperbolic sine is defined as

$\displaystyle{\sinh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{2}}$

Applying this formula, the expression becomes

$\displaystyle=\frac{{{{e}}^{{{3}}}-{{e}}^{{-{3}}}}}{{2}}$

$\displaystyle=\frac{{{e}}^{{3}}}{{2}}-\frac{{{e}}^{{-{3}}}}{{2}}$

Expressing it with positive exponent only, it turns into

$\displaystyle=\frac{{{e}}^{{3}}}{{2}}-\frac{{1}}{{{2}{{e}}^{{3}}}}$

This is the exact value of sinh(3).

Its approximate value is

$\displaystyle\approx{10.01787492}$

Rounding off to three decimal places, it becomes

$\displaystyle\approx{10.018}$

Therefore, $\displaystyle{\sinh{{\left({3}\right)}}}\approx{10.018}$ .

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