Stationary points using trigonometry

5 The equation of a curve is \(y = 2 \cos x + \sin 2 x\). Find the \(x\)-coordinates of the stationary points on the curve for which \(0 < x < \pi\), and determine the nature of each of these stationary points.

6 The equation of a curve is \(y = 3 \sin x + 4 \cos ^ { 3 } x\).

1. Find the \(x\)-coordinates of the stationary points of the curve in the interval \(0 < x < \pi\).
2. Determine the nature of the stationary point in this interval for which \(x\) is least.

10. (a) Use the identity for \(\sin ( A + B )\) to prove that $$\sin 2 A \equiv 2 \sin A \cos A$$ (b) Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ \ln \left( \tan \left( \frac { 1 } { 2 } x \right) \right) \right] = \operatorname { cosec } x$$ A curve \(C\) has the equation $$y = \ln \left( \tan \left( \frac { 1 } { 2 } x \right) \right) - 3 \sin x , \quad 0 < x < \pi$$ (c) Find the \(x\) coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give your answers to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)