Calculate cosh (x+y) given sinh x = 4/3 and sinh y = 3/4

Find the value of $\cosh (x+y)$ given that $\sinh x = \frac{4}{3}$ and $\sinh y = \frac{3}{4}$ .

We use the identity $\cosh^2 = 1 + \sinh^2$ in both cases.

$\begin{align*} \cosh^2 x &= 1 + \sinh^2 x &\implies && \cosh^2 x &= 1 + \frac{16}{9} & \implies && \cosh x &= \frac{5}{3} \\ \cosh^2 y &= 1 + \sinh^2 y &\implies && \cosh^2 y &= 1 + \frac{9}{16} & \implies && \cosh y &= \frac{5}{4}. \end{align*}$

Then we recall (Section 6.19, Exercise #6) the formula for $\cosh (x+y)$ ,

$\cosh (x+y) = \cosh x \cosh y + \sinh x \sinh y.$

Therefore, we can compute

$\begin{align*} \cosh (x+y) &= \cosh x \cosh y + \sinh x \sinh y \\ &= \frac{5}{3} \cdot \frac{5}{4} + \frac{4}{3}\cdot \frac{3}{4} \\ &= \frac{25}{12} + 1 \\ &= \frac{37}{12}. \end{align*}$

Stumbling Robot

Calculate cosh (x+y) given sinh x = 4/3 and sinh y = 3/4

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