Factorization method

3 Solve, by factorising, the equation $$6 \cos \theta \tan \theta - 3 \cos \theta + 4 \tan \theta - 2 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

5 Jessica, a maths student, is asked by her teacher to solve the equation \(\tan x = \sin x\), giving all solutions in the range \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) The steps of Jessica's working are shown below. $$\begin{aligned} & \tan x = \sin x \\ & \text { Step } 1 \Rightarrow \frac { \sin x } { \cos x } = \sin x \quad \text { Write } \tan x \text { as } \frac { \sin x } { \cos x } \\ & \text { Step } 2 \Rightarrow \sin x = \sin x \cos x \quad \text { Multiply by } \cos x \\ & \text { Step } 3 \Rightarrow 1 = \cos x \quad \text { Cancel } \sin x \\ & \Rightarrow \quad x = 0 ^ { \circ } \text { or } 360 ^ { \circ } \end{aligned}$$ The teacher tells Jessica that she has not found all the solutions because of a mistake.
Explain why Jessica's method is not correct.
[0pt] [2 marks]

2. Some A level students were given the following question. Solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation $$\cos \theta = 2 \sin \theta$$ The attempts of two of the students are shown below. Student \(B\) $$\begin{aligned} \cos \theta & = 2 \sin \theta \\ \cos ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta \\ 1 - \sin ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta \\ \sin ^ { 2 } \theta & = \frac { 1 } { 5 } \\ \sin \theta & = \pm \frac { 1 } { \sqrt { 5 } } \\ \theta & = \pm 26.6 ^ { \circ } \end{aligned}$$

1. Identify an error made by student \(A\). Student \(B\) gives \(\theta = - 26.6 ^ { \circ }\) as one of the answers to \(\cos \theta = 2 \sin \theta\).
2. 1. Explain why this answer is incorrect.
   2. Explain how this incorrect answer arose.

7 It is given that \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\).

1. Find the possible values of \(\tan \theta\).
2. Hence solve the equation \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\), giving all solutions for \(\theta\), in degrees, in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).