A mechanical linkage tracks an angular displacement θ \theta\,θ radians. Its extension, EEE, is modelled by
E(θ)=−2cosec(θ+π)E(\theta) = -2\operatorname{cosec}(\theta + \pi)E(θ)=−2cosec(θ+π) for 0≤θ≤2π0 \leq \theta \leq 2\pi0≤θ≤2π
Show that E(θ)=2cosecθE(\theta) = 2\operatorname{cosec}\thetaE(θ)=2cosecθ.
Sketch the graph of E E\,E against θ \theta\,θ for 0≤θ≤2π0 \leq \theta \leq 2\pi0≤θ≤2π, stating the equations of any vertical asymptotes and the coordinates of any local turning points.