A robotic welding head follows a path C C\,C in the xyxyxy-plane described by the parametric equations
x=2sinθ+5cosθ,y=4cos2θ+2sinθ,0≤θ≤π x = 2\sin \theta + 5\cos \theta, \quad y = 4\cos^2 \theta + 2\sin \theta, \quad 0 \le \theta \le \pi x=2sinθ+5cosθ,y=4cos2θ+2sinθ,0≤θ≤πShow that dydx=1\displaystyle \frac{dy}{dx} = 1dxdy=1 where θ=0\theta = 0θ=0.
The point P P\,P lies on C C\,C where θ=0\theta = 0θ=0.
Find the equation of the tangent to the path C C\,C at the point PPP, giving your answer in the form y=mx+cy = mx + cy=mx+c.
The tangent to the path at P P\,P intersects the curve C C\,C again at the point QQQ.
Show that the value of θ \theta\,θ at point Q Q\,Q satisfies the equation
4cos2θ−5cosθ+1=0 4\cos^2 \theta - 5\cos \theta + 1 = 0 4cos2θ−5cosθ+1=0Hence find the exact value of the yyy-coordinate of QQQ.
