A robotic surveillance arm is anchored at a fixed pivot O O\,O at the origin of a coordinate system. The arm consists of two segments, OP OP\,OP and PQPQPQ, with lengths 2k 2k\,2k and k k\,k respectively. A sensor is attached at point QQQ.
The configuration of the arm is defined by an angle ϕ \phi\,ϕ in the range 0≤ϕ≤π0 \le \phi \le \pi0≤ϕ≤π. The coordinates of P P\,P are (2kcosϕ,2ksinϕ)(2k \cos \phi, 2k \sin \phi)(2kcosϕ,2ksinϕ) and the position of Q Q\,Q relative to P P\,P is (kcos2ϕ,ksin2ϕ)(k \cos 2\phi, k \sin 2\phi)(kcos2ϕ,ksin2ϕ).
Show that the xxx-coordinate of the sensor at Q Q\,Q is given by
x=k(2cosϕ+cos2ϕ) x = k(2\cos \phi + \cos 2\phi) x=k(2cosϕ+cos2ϕ)Using a double angle identity, show that
x=k(2cos2ϕ+2cosϕ−1) x = k(2\cos^2 \phi + 2\cos \phi - 1) x=k(2cos2ϕ+2cosϕ−1)The expression in part (b) can be written as x=2k(cosϕ+12)2−3k2\displaystyle x = 2k\left(\cos \phi + \frac{1}{2}\right)^2 - \frac{3k}{2}x=2k(cosϕ+21)2−23k. Find the value of ϕ \phi\,ϕ for which x x\,x is a minimum and state this minimum value in terms of kkk.
Calculate the exact distance OQ OQ\,OQ when x x\,x is at its minimum value, giving your answer in terms of kkk.
