Mixed sin and cos linear

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 $$3 \tan x - 10 \cos x = 10$$

Solve the equation $$\cos(x + 30°) = 2\cos x,$$ giving all solutions in the interval \(-180° < x < 180°\). [5]

Showing all necessary working, solve the equation \(\sin(\theta - 30°) + \cos \theta = 2 \sin \theta\), for \(0° < \theta < 180°\). [4]

Given the equation \(\sin(x + 45°) = 2\cos x\), show that \(\sin x + \cos x = 2\sqrt{2}\cos x\). Hence solve, correct to 2 decimal places, the equation for \(0° < x < 360°\). [6]

The angle \(\theta\) satisfies the equation \(\sin(\theta + 45°) = \cos\theta\).

1. Using the exact values of \(\sin 45°\) and \(\cos 45°\), show that \(\tan\theta = \sqrt{2} - 1\). [5]
2. Find the values of \(\theta\) for \(0° < \theta < 360°\). [2]