Factorization method

1. (a) Show that the equation

$$\tan 2 x = 5 \sin 2 x$$ can be written in the form $$( 1 - 5 \cos 2 x ) \sin 2 x = 0$$ (b) Hence solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), $$\tan 2 x = 5 \sin 2 x$$ giving your answers to 1 decimal place where appropriate.
You must show clearly how you obtained your answers.

\includegraphics{figure_6} The diagram shows part of the curve with equation $$y = 4\sin^2 x + 8\sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\). [3]
2. Show that the equation of the curve can be written $$y = 5 + 8\sin x - 2\cos 2x,$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]

Solve the following equation, for \(0 \leq x \leq \pi\), giving your answers in terms of \(\pi\). $$\sin 5x - \cos 5x = \cos x - \sin x.$$ [8]

Solve the equation $$\cos 2\theta - \cos 4\theta = \sin 3\theta \quad \text{for} \quad 0 \leq \theta \leq \pi$$ [6]

Find all the roots of the equation $$\cos \theta + \cos 3\theta + \cos 5\theta = 0$$ lying in the interval \([0, \pi]\). Give all the roots in radians in terms of \(\pi\). [8]

1. Factorise \(8xy - 4x + 6y - 3\) into the form \((ax + b)(cy + d)\) where \(a, b, c\) and \(d\) are integers
2. Hence, or otherwise, solve $$8\sin(x^2)\cos\left(e^{\frac{x}{3}}\right) - 4\sin(x^2) + 6\cos\left(e^{\frac{x}{3}}\right) - 3 = 0$$ where \(0° < x < 19°\), giving your answers to 1 decimal place.

[6 marks]