Express 8sinx−15cosx8 \sin x - 15 \cos x8sinx−15cosx in the form Rsin(x−α)R \sin(x - \alpha)Rsin(x−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 function ggg is defined by
g(θ)=15+8sin(3θ−π4)−15cos(3θ−π4),θ>0 g(\theta) = 15 + 8 \sin\left(3\theta - \frac{\pi}{4}\right) - 15 \cos\left(3\theta - \frac{\pi}{4}\right), \quad \theta > 0 g(θ)=15+8sin(3θ−4π)−15cos(3θ−4π),θ>0Find (i) the minimum value of g(θ)g(\theta)g(θ) (ii) the smallest value of θ\thetaθ at which this minimum value occurs.
The function hhh is defined by
h(β)=5−(8sinβ−15cosβ)2 h(\beta) = 5 - (8 \sin \beta - 15 \cos \beta)^2 h(β)=5−(8sinβ−15cosβ)2Find the range of hhh.
