Question 5 - A-Level Maths

Using the definition $\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)$, prove the identity $$4 \sinh ^ { 3 } \theta + 3 \sinh \theta = \sinh 3 \theta$$
Given that $x = \sinh \theta$ and $16 x ^ { 3 } + 12 x - 3 = 0$, find the value of $\theta$ in terms of a natural logarithm.
Hence find the real root of the equation $16 x ^ { 3 } + 12 x - 3 = 0$, giving your answer in the form $2 ^ { p } - 2 ^ { q }$, where $p$ and $q$ are rational numbers.
[0pt] [2 marks]