Question

(a) To determine the numerical value of given expression.

(b) To determine the numerical value of given expression.

Short answer

(a) The value of \(\sin h1 = 1\).

(b) The value of \(\sin {h^{ - 1}}1 = 0.881\)

Step-by-step solution

Given expressions

(a) The expression is \(\sin h1\).

(b) The expression is \(\sin {h^{ - 1}}1\).

Definition of hyperbolic function

The definition of the inverse hyperbolic sine function is\(sinhx = \frac{{{e^x} - {e^{ - x}}}}{2}\).

The definition of the inverse hyperbolic sine function is\(sin{h^{ - 1}}x = ln\left( {x + \sqrt {{x^2} + 1} } \right),x\^I i\).

Use the formula and substitute values

(a)

The definition of the hyperbolic functions, \(\sin hx = \frac{{{e^2} - {e^{ - x}}}}{2}\).

Substitute 1 for\(x\).

\(\begin{array}{c}\sin h1 = \frac{{{e^1} - {e^{ - 1}}}}{2}\\ = \frac{{e - \frac{1}{e}}}{{2e}}\\ = \frac{{{e^2} - 1}}{{2e}}\\ \approx \frac{{7.389 - 1}}{{2(2.718)}}\end{array}\)

Simplify further and obtain the value of \(\sin h1\).

\(\sin h1 = \frac{{6.389}}{{5.436}} \approx 1.175\)

Thus, the value of \(\sin h1 \approx 1.175\).

Use the formula and substitute values

(b)

The definition of the inverse hyperbolic sine function is, \(\sin {h^{ - 1}}x = \ln \left( {x + \sqrt {{x^2} + 1} } \right),x \in i\).

Substitute 1 for \(x\).

\(\begin{array}{c}\sin {h^{ - 1}}1 = \ln \left( {1 + \sqrt {{1^2} + 1} } \right)\\ = \ln (1 + \sqrt 2 )\\ = 0.881\end{array}\)

Thus, the value of \(\sin {h^{ - 1}}1 = 0.881\).