Hyperbolic Functions 
Hey everyone, I need help with questions on hyperbolic functions.
I was able to do part (a). 
I proved for $\sinh(3y)$ by doing this:
\begin{align*}
\sinh(3y) &= \sinh(2y +y)\\
&= \sinh(2y)\cosh(y) + \cosh(2y)\sinh(y)\\
&= 2\sinh(y)\cosh(y)\cosh(y) + (\cosh^2(y)+\sinh^2(y))\sinh(y)\\
&= 2\sinh(y)(1+\sinh^2(y)) + (1+\sinh^2(y) + \sinh^2(y))\sinh(y)\\
&= 2\sinh(y) + 2\sinh^3(y) + \sinh(y) +2\sinh^3(y)\\
&= 4\sinh^3(y) + 3\sinh(y).
\end{align*}
Therefore, $0 = 4\sinh^3(y) + 3\sinh(y) - \sinh(3y)$.
I have no clue what to do for part (b) and part (c) but I do see similarities between part (a) and part(b) as you can subtitute $x = \sinh(y)$.
But yeah, I'm stuck and help would be very much appreciated.

In the identity that you proved, put $\sinh 3y=2$. Then if $x=\sinh y$, the identity says that $4x^3+3x-2=0$. So we are almost finished, we have shown this $x$ is a solution of the equation.
Note that $\sinh t$ is a strictly increasing function, and that $\sinh t$ is large negative when $t$ is large negative, and large positive when $t$ is large positive.  So it has an inverse function $\sinh^{-1}$, defined everywhere. We have $\sinh 3y=2$ if and only if $3y=\sinh^{-1} 2$ if and only if $y=\frac{1}{3}\sinh^{1}2$.  Finally, to get $x$, take the $\sinh$ of this.
Now start from $ax^3+bx+c=0$.  We want to manipulate this equation so that it will look like $4t^3+3t-d=0$, so that we can use the same trick. 
We will make the substitution $x=kt$ for some constant $k$. This yields the equation $ak^3t^3+bkt +c=0$. We want the lead coefficient to be $4$. So multiply through by $\frac{4}{ak^3}$. We get the equation
$$4t^3+\frac{4b}{ak^2}t+\frac{4b}{ak^3}c=0.\tag{$1$}$$
We want the coefficient of $t$ to be $3$. So we need $\frac{4b}{ak^2}=3$. That gives 
$$k=2\sqrt{\frac{b}{3a}}.$$
To solve the cubic $(1)$ using the $\sinh$ method, let $\sinh 3y=-\frac{4b}{ak^3}c$, where $k$ is as just calculated.
