Solve the equation \[ \cos^{-1}(x+\tfrac{1}{2}) + \cos^{-1}x + \cos^{-1}(x-\tfrac{1}{2}) = \frac{3\pi}{2}. \]
A kite of weight \(w\) is in the form of a circular sector \(AOB\) of angle 60\(^\circ\) at \(O\). The centre of gravity is at the mid-point of the central radius \(OC\); and the string, which is horizontal near the kite, passes through the point of intersection of \(OC\) and the chord \(AB\). Shew that if the effect of the wind is a uniform pressure normal to the face of the kite, the kite will make an angle \(\theta\) with the horizontal, where \(\sin\theta = \sqrt{\frac{4-\pi}{\pi(\sqrt{3}-1)}}\). Find the tension in the string.
Three smooth equal cylinders of radius \(a\) and weight \(w\) have their axes parallel and horizontal. Two of them rest on a smooth horizontal table; the third rests between them and equilibrium is maintained by means of a string of length \((2\pi a + l)\) passing round all three and everywhere perpendicular to the direction of their axes. If \(l<8a\), shew that the tension in the string is \(\frac{w(l-4a)}{2\{l(8a-l)\}^{\frac{1}{2}}}\).
The upper half of a rectangular window is of width \(2a\) and height \(2b\). It fits loosely in its frame and is just supported by two light inextensible sashcords passing over light frictionless pulleys and carrying equal weights. One of these cords breaks. Shew that the window will fall provided \(\mu < \frac{b}{a}\), where \(\mu\) is the coefficient of friction between the window and its frame. Find the acceleration with which it descends.
A small rectangular target can be rotated about one edge, kept horizontal, and makes an angle \(\phi\) with the horizontal plane. It is fired at by a gun standing at the same height and at distance \(r\), having a fixed muzzle velocity \(u\), where \(u^2 > rg\). Of the two possible angles of elevation of the gun shew that the less is the more effective (in that it gives hits with a greater error in laying for elevation) for all values of \(\phi\) other than those of an interval equal to \(\tan^{-1}\left(\frac{u^2}{rg}\right)\). (The shell is to be treated as a particle in vacuo.)
A particle of mass \(m\) lies upon a smooth horizontal table. To it is fastened a light inextensible string which passes through a small smooth hole in the table at distance \(a\) from the particle, and carries a second particle of mass \(M\) hanging freely. The first particle is set in motion with a velocity \(V\) perpendicular to the string which is just sufficient to maintain \(M\) in equilibrium. Establish the stability of this equilibrium by shewing that if \(M\) is drawn downwards through a small distance \(x\), and then let go, it will commence to ascend with acceleration \(3Mgx/a(M+m)\).
Find the value of \(\tan \frac{\pi}{16}\) without using tables. If \(\alpha, \beta\) are values of \(\theta\), not differing by a multiple of \(\pi\), which satisfy the equation \(a\cos\theta+b\sin\theta=c\), prove that \(\tan\alpha+\tan\beta=\frac{2ab}{c^2-b^2}\); also find \(\tan 2\alpha+\tan 2\beta\).
Prove that
In a triangle \(ABC\) prove that if \(P\) is the orthocentre and \(O\) the circumcentre \[ PO^2 = R^2(1-8\cos A\cos B\cos C). \] If \(N\) is the ninepoint centre, prove that \(NA^2+NB^2+NC^2+NO^2=3R^2\).
If \(\sin(\xi+i\eta) = x \sin\alpha\) where \(x > 1\), find how \(\xi\) and \(\eta\) vary as \(\alpha\) varies from \(0\) to \(\pi\). Find the sum to infinity of the series \(\cos\alpha\cos\beta+\frac{1}{2}\cos^2\alpha\cos 2\beta+\frac{1}{3}\cos^3\alpha\cos 3\beta+\dots\).