Question

What is the value of \[\sinh\left( {3z} \right)\]?
A. \[2\sinh z – 4\sinh^{3}z\]
B. \[4sinh^{3}z - 3\sinh z\]
C. \[3\sinh z + 4sinh^{3}z\]
D. None of these

Answer 1

Hint: In the given question, hyperbolic sine function is given. By using the various formulas of the hyperbolic sine function, we will find the value of \[\sinh\left( {3z} \right)\].

Formula Used:
\[\sinh(A + B) = \sinh A \cosh B + \cosh A\sinh B\]
\[\sinh2A = 2\sinh A \cosh A\]
\[\cosh 2A = cos{h^2}A + \sinh^{2}A\]
\[\cosh^{2}A = 1 + \sinh^{2}A\]

Complete step by step solution:
The given hyperbolic function is \[\sinh\left( {3z} \right)\].
Let’s simplify the above function.
\[\sinh\left( {3z} \right) = \sinh\left( {2z + z} \right)\]
Now apply the formula \[\sinh(A + B) = \sinh A\cosh B + \cosh A\sinh B\] on the right-hand side of the above equation.
\[ \Rightarrow \]\[\sinh(3z) = \sinh\left( {2z} \right)\cosh\left( z \right) + \cosh\left( {2z} \right)\sinh\left( z \right)\]
\[ \Rightarrow \]\[\sinh(3z) = \left[ {2\sinh\left( z \right)\cosh\left( z \right)} \right]\cosh\left( z \right) + \left[ {\cosh^{2}\left( z \right) + \sinh^{2}\left( z \right)} \right]\sinh\left( z \right)\] [since \[\sinh 2A = 2\sinh A\cosh A\] and \[\cosh 2A = \cosh^{2}A + \sinh^{2}A\]]

Simplify the above equation.
\[\sinh(3z) = 2\sinh\left( z \right)\cosh^{2}\left( z \right) + \left[ {\cosh^{2}\left( z \right) + \sinh^{2}\left( z \right)} \right]\sinh\left( z \right)\]
\[ \Rightarrow \]\[\sinh(3z) = 2\sinh\left( z \right)\left[ {1 + \sinh^{2}\left( z \right)} \right] + \left[ {1 + \sinh^{2}\left( z \right) + \sinh^{2}\left( z \right)} \right]\sinh\left( z \right)\] [Since \[\cosh^{2}A = 1 + sinh^{2}A\]]
\[ \Rightarrow \]\[\sinh(3z) = 2\sinh\left( z \right) + \sinh^{3}\left( z \right) + \left[ {1 + 2sinh^{2}\left( z \right)} \right]\sinh\left( z \right)\]
\[ \Rightarrow \]\[\sinh(3z) = 2\sinh\left( z \right) + 2\sinh^{3}\left( z \right) + \sinh\left( z \right) + 2\sinh^{3}\left( z \right)\]
\[\Rightarrow \]\[\sinh(3z) = 3\sinh\left( z \right) + 4\sinh^{3}\left( z \right)\]

Hence the correct option is option C.
Note: Students are often get confused between the basic trigonometric formulas and hyperbolic trigonometric formulas.
In basic trigonometry \[\cos^{2}A = 1 -\ sin^{2}A\]. But in hyperbolic trigonometry \[\cosh^{2}A = 1 + sinh^{2}A\].