Convert to quadratic in tan

11

1. Solve the equation \(3 \tan ^ { 2 } x - 5 \tan x - 2 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
2. Find the set of values of \(k\) for which the equation \(3 \tan ^ { 2 } x - 5 \tan x + k = 0\) has no solutions.
3. For the equation \(3 \tan ^ { 2 } x - 5 \tan x + k = 0\), state the value of \(k\) for which there are three solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), and find these solutions.

4

1. Show that the equation $$\sin x - \cos x = \frac { 6 \cos x } { \tan x }$$ can be expressed in the form $$\tan ^ { 2 } x - \tan x - 6 = 0 .$$
2. Hence solve the equation \(\sin x - \cos x = \frac { 6 \cos x } { \tan x }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

9

1. Solve the equation \(\tan x = - 3\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
2. 1. Given that $$7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6$$ show that $$\tan ^ { 2 } \theta + \tan \theta - 6 = 0$$
   2. Hence solve the equation \(7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
      (4 marks)

5. (a) Given that $$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0 .$$ (b) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0 .$$

3

1. Solve the equation \(\tan x = - \frac { 1 } { 3 }\), giving all the values of \(x\) in the interval \(0 < x < 2 \pi\) in radians to two decimal places.
2. Show that the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ can be written in the form \(3 \tan ^ { 2 } x - 5 \tan x - 2 = 0\).
3. Hence, or otherwise, solve the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ giving all the values of \(x\) in the interval \(0 < x < 2 \pi\) in radians to two decimal places.
   (4 marks)

4

1. Find the possible values of \(\tan \theta\), giving your answers in exact form.
   4
2. Hence, or otherwise, solve the equation $$5 \cos ^ { 2 } \theta - 4 \sin ^ { 2 } \theta = 0$$ giving all solutions of \(\theta\) to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) \begin{center} \begin{tabular}{|l|l|} \hline