It is given that

$f(θ)=5cos⁡θ+sin⁡θf(\theta) = 5\cos\theta + \sin\theta$

and that $f(θ)=Rcos⁡(θ−α)f(\theta) = R\cos(\theta - \alpha)$ , where $0\,$ and $0≤α≤π2\displaystyle 0 \leq \alpha \leq \frac{\pi}{2}$ .

Find the value of $R\,$ and the value of $α\alpha$ , each to three decimal places.

Hence solve, for $\leq \theta \leq 2\pi$ , the equation

$5cos⁡θ+sin⁡θ=25\cos\theta + \sin\theta = 2$

Give your answers to three decimal places.

Find the minimum value of

$5cos⁡4x+sin⁡4x+155\cos 4x + \sin 4x + 15$

and the smallest positive value of $x\,$ at which this minimum occurs. Give each answer to three decimal places.