Question

Find the possible values of $\sin hx$ for which
$12\cos h^{2}x+7\sin hx=24$
(You may find the identity $\cos h^{2}x-\sin h^{2}x=1$ useful)
Hence find the possible values of $x$, leaving your answers as natural logarithms

Answer 1

Using the identity ${\cosh }^{2}x-{\sinh }^{2}x=1$

Let $\sinh x=t$

$\Rightarrow 12(1+t^2)+7t=24$

$\Rightarrow 12t^2+7t-12=0$

$\Rightarrow (3t+4)(4t-3)=0$

$\Rightarrow t=-\frac{4}{3}$ or $t=\frac{3}{4}$

Case $1$: $t=-\frac{4}{3}$

$\Rightarrow \sinh x=-\frac{4}{3}$

$\Rightarrow \frac{e^x-e^{-x}}{2}=-\frac{4}{3}$

$\Rightarrow (3e^x-1)(e^x+3)=0$

$\Rightarrow e^x=\frac{1}{3}$ (as $e^x$ is always positive)

$\Rightarrow x=\frac{1}{3}\ln 3$

Case $2$: $t=\frac{3}{4}$

$\Rightarrow (2e^x+1)(e^x-2)=0$

$\Rightarrow e^x=2$

$\Rightarrow x=\ln 2$