Solve $4\sinh (x)+3\cosh (x)=0$ for $x$

Question

Solve the following for $x$ giving your answer to $3$ significant figures: $$4\sinh (x)+3\cosh (x)=0$$

I need help understanding hyperbolic functions.

Answer 1

by Definition we have $$4\left(\frac{e^x-e^{-x}}{2}\right)+3\left(\frac{e^x+e^{-x}}{2}\right)=0$$ multiplying by $2$ and rearanging we get $$7e^x-e^{-x}=0$$ multiplying by $e^x$ we get $$7e^{2x}-1=0$$ and from here we get $$e^{2x}=\frac{1}{7}$$ taking the logarithm we have $$x=\frac{1}{2}\ln\left(\frac{1}{7}\right)$$

Answer 2

Substitute $t=e^x$. Then

$4 \sinh (x)+3 \cosh (x)=0$

iff

$4(t-\frac{1}{t})+3(t+\frac{1}{t})=0$

iff

$7t^2=1$.

Your turn !