Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation,
$$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$
giving your answers to one decimal place.
(a) A student's attempt at the question
"Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below.
$$\begin{gathered}
7 \tan x = 8 \sin x
7 \times \frac { \sin x } { \cos x } = 8 \sin x
7 \sin x = 8 \sin x \cos x
7 = 8 \cos x
\cos x = \frac { 7 } { 8 }
x = 29.0 ^ { \circ } \text { (to } 3 \mathrm { sf } \text { ) }
\end{gathered}$$
Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation
$$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$