Mixed sin and cos linear

2 Find all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) which satisfy the equation \(\sin 3 x + 2 \cos 3 x = 0\).

1

1. Show that the equation $$3 ( 2 \sin x - \cos x ) = 2 ( \sin x - 3 \cos x )$$ can be written in the form \(\tan x = - \frac { 3 } { 4 }\).
2. Solve the equation \(3 ( 2 \sin x - \cos x ) = 2 ( \sin x - 3 \cos x )\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Solve, for \(- 180 ^ { \circ } < \theta \leqslant 180 ^ { \circ }\), the equation $$3 \tan \left( \theta + 43 ^ { \circ } \right) = 2 \cos \left( \theta + 43 ^ { \circ } \right)$$

6. (a) Given that \(\sin \theta = 5 \cos \theta\), find the value of \(\tan \theta\).
(b) Hence, or otherwise, find the values of \(\theta\) in the interval \(0 \leqslant \theta < 360 ^ { \circ }\) for which $$\sin \theta = 5 \cos \theta ,$$ giving your answers to 1 decimal place.

1. (a) Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that the equation

$$3 \cos x - 2 \sin x = 1$$ can be written in the form $$2 t ^ { 2 } + 2 t - 1 = 0$$ (b) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$3 \cos x - 2 \sin x = 1$$ giving your answers to one decimal place.

1. (a) Given that \(t = \tan \frac { X } { 2 }\) prove that

$$\cos x \equiv \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } }$$ (b) Show that the equation $$3 \tan x - 10 \cos x = 10$$ can be written in the form $$( t + 2 ) \left( a t ^ { 2 } + b t + c \right) = 0$$ where \(t = \tan \frac { X } { 2 }\) and \(a , b\) and \(c\) are integers to be determined.
(c) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$3 \tan x - 10 \cos x = 10$$