Quadratic in sin²/cos²/tan²

3 Solve the equation \(3 \tan ^ { 2 } \theta + 1 = \frac { 2 } { \tan ^ { 2 } \theta }\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 Solve the equation \(4 \sin ^ { 4 } \theta + 12 \sin ^ { 2 } \theta - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

5

1. Show that the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) can be written in the form $$2 \sin ^ { 4 } \theta + \sin ^ { 2 } \theta - 1 = 0 .$$
2. Hence solve the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

2

1. Show that the equation \(4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta\) may be written in the form \(4 x ^ { 2 } + 7 x - 2 = 0\), where \(x = \sin ^ { 2 } \theta\).
2. Hence solve the equation \(4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

7 Showing your method clearly, solve the equation \(4 \sin ^ { 2 } \theta = 3 + \cos ^ { 2 } \theta\), for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

3. Giving your answers in terms of \(\pi\), solve the equation $$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).

1. Giving your answers in terms of \(\pi\), solve the equation

$$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).

4 In this question you must show detailed reasoning.
Solve the equation \(3 \sin ^ { 4 } \phi + \sin ^ { 2 } \phi = 4\), for \(0 \leqslant \phi < 2 \pi\), where \(\phi\) is measured in radians.

4 Solve the equation \(\tan ^ { 2 } 2 \theta - 3 = 0\) giving all the solutions for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) [0pt] [4 marks] \(5 \quad \mathrm { f } ^ { \prime } ( x ) = \left( 2 x - \frac { 3 } { x } \right) ^ { 2 }\) and \(\mathrm { f } ( 3 ) = 2\) Find \(\mathrm { f } ( x )\).
[0pt] [4 marks]

5 Solve the equation $$\sin ^ { 2 } x = 1$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\) [0pt] [3 marks]