Let f(θ)=3cosθ−6sinθf(\theta) = 3\cos\theta - 6\sin\thetaf(θ)=3cosθ−6sinθ for θ∈R\theta \in \mathbb{R}θ∈R.
Express f(θ)f(\theta)f(θ) in the form Rcos(θ+α)R\cos(\theta + \alpha)Rcos(θ+α), where R>0R > 0R>0 and 0<α<π20 < \alpha < \frac{\pi}{2}0<α<2π. Give the exact value of RRR and the value of α\alphaα, in radians, to 3 decimal places.
The curve with equation y=cosθy = \cos\thetay=cosθ is transformed onto the curve with equation y=f(θ)y = f(\theta)y=f(θ) by a sequence of two transformations.
Given that the first transformation is a stretch and the second is a translation:
(i) Describe fully the transformation that is a stretch. (ii) Describe fully the transformation that is a translation.
Given g(θ)=755+(f(θ))2g(\theta) = \frac{75}{5 + (f(\theta))^2}g(θ)=5+(f(θ))275, find the range of ggg.
