A robot arm's adjustment angle, θ\thetaθ (in radians), is determined by the equilibrium condition
H(θ)=2tanθ+5θ−4,0<θ<π2 H(\theta) = 2\tan \theta + 5\theta - 4, \quad 0 < \theta < \frac{\pi}{2} H(θ)=2tanθ+5θ−4,0<θ<2πThe equation H(θ)=0H(\theta) = 0H(θ)=0 has a single root α\alphaα.
Show that 0.5<α<0.60.5 < \alpha < 0.60.5<α<0.6.
Show that α\alphaα is a solution of the equation
θ=0.8−0.4tanθ \theta = 0.8 - 0.4\tan \theta θ=0.8−0.4tanθThe iterative formula
θn+1=0.8−0.4tanθn \theta_{n+1} = 0.8 - 0.4\tan \theta_n θn+1=0.8−0.4tanθnis used to find α\alphaα.
Starting with θ1=0.55\theta_1 = 0.55θ1=0.55 and using the iterative formula, (i) find, to 4 decimal places, the value of θ2\theta_2θ2. (ii) find, to 4 decimal places, the value of α\alphaα.
