The horizontal position x x\,x of a piston in a high-precision engine is modelled by the equation
x=18cos2(2θ)0<θ<π4 x = 18 \cos^2(2\theta) \qquad 0 < \theta < \frac{\pi}{4} x=18cos2(2θ)0<θ<4πwhere θ \theta\,θ is the crankshaft angle in radians.
Show that the rate of change of the angle with respect to the position, dθdx\displaystyle \frac{d\theta}{dx}dxdθ, can be expressed in the form
dθdx=−1ABx−x2 \frac{d\theta}{dx} = -\frac{1}{A\sqrt{Bx - x^2}} dxdθ=−ABx−x21where A A\,A and B B\,B are integers to be determined.
