Rational trig expressions

5 Solve the equation $$\frac { \tan \theta + 3 \sin \theta + 2 } { \tan \theta - 3 \sin \theta + 1 } = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\).

3 Solve the equation \(\frac { \tan \theta + 2 \sin \theta } { \tan \theta - 2 \sin \theta } = 3\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

1 Solve the equation \(2 \cos \theta = 7 - \frac { 3 } { \cos \theta }\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

3 Solve the equation \(\frac { 13 \sin ^ { 2 } \theta } { 2 + \cos \theta } + \cos \theta = 2\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

17 In this question you must show detailed reasoning. Solve the equation \(2 \sin x + \sec x = 4 \cos x\), where \(- \pi < x < \pi\).

6 In this question you must show detailed reasoning. Solve the equation \(\tan x - 3 \cot x = 2\) for values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).