Identify student error in trig solution

1. In this question you should show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Solve, for \(0 < \theta \leqslant 450 ^ { \circ }\), the equation $$5 \cos ^ { 2 } \theta = 6 \sin \theta$$ giving your answers to one decimal place.
2. (a) A student's attempt to solve the question
   "Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(3 \tan x - 5 \sin x = 0\) " is set out below. $$\begin{gathered} 3 \tan x - 5 \sin x = 0 \\ 3 \frac { \sin x } { \cos x } - 5 \sin x = 0 \\ 3 \sin x - 5 \sin x \cos x = 0 \\ 3 - 5 \cos x = 0 \\ \cos x = \frac { 3 } { 5 } \\ x = 53.1 ^ { \circ } \end{gathered}$$ Identify two errors or omissions made by this student, giving a brief explanation of each. The first four positive solutions, in order of size, of the equation $$\cos \left( 5 \alpha + 40 ^ { \circ } \right) = \frac { 3 } { 5 }$$ are \(\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }\) and \(\alpha _ { 4 }\) (b) Find, to the nearest degree, the value of \(\alpha _ { 4 }\)

1. (i) 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.
(ii) (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. $$\begin{gathered} 7 \tan x = 8 \sin x \\ 7 \times \frac { \sin x } { \cos x } = 8 \sin x \\ 7 \sin x = 8 \sin x \cos x \\ 7 = 8 \cos x \\ \cos x = \frac { 7 } { 8 } \\ x = 29.0 ^ { \circ } \text { (to } 3 \text { sf) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$