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 \(\cos\alpha\cos\beta+\frac{1}{2}\cos^2\alpha\cos 2\beta+\frac{1}{3}\cos^3\alpha\cos 3\beta+\dots\).
Find the conditions of equilibrium of a system of coplanar forces acting on a body. A uniform rod of length \(l\) rests in equilibrium, partly in and partly outside a smooth hemispherical bowl of radius \(a\) whose rim is horizontal. Shew that the inclination \(\theta\) of the rod to the horizontal is determined by \(l\cos\theta=4a\cos 2\theta\). Shew also that \(l\) must be less than \(4a\) and greater than \(2a\sqrt{2}/\sqrt{3}\).
Three equal uniform rods of length \(l\) and weight \(w\) are smoothly jointed together to form a triangle \(ABC\). This triangle is hung up by the joint \(A\), and by two strings each of length \(l/\sqrt{2}\) a weight \(W\) is attached to \(B\) and \(C\). The system hangs under gravity. Shew that the stress in the rod \(BC\) is \(\frac{1}{\sqrt{3}}\{W+\frac{w}{2}(1+\sqrt{3})\}\).
A heavy uniform rod \(AB\) of weight \(W\) rests with one end \(A\) on a rough horizontal plane and the other end against an equally rough vertical wall. The vertical plane through the rod is perpendicular to the wall. Shew that if the rod makes an angle \(\alpha\) with the horizontal it cannot be in equilibrium unless \(\alpha\) is greater than \(\pi/2-2\epsilon\), where \(\epsilon\) is the angle of friction. If \(\alpha\) be less than \(\pi/2-2\epsilon\), shew that the magnitude of the least force which acting on \(A\) in the vertical plane through the rod will just maintain equilibrium is \(W\cos(\alpha+2\epsilon)/2\sin(\alpha+\epsilon)\), and find the direction of the force.
Shew that in general there are two directions in which a particle can be projected under gravity with a given velocity from a given point \(P\) so as to pass through another given point \(Q\). Prove that the differences of the tangents of the angles the directions of motion at \(P\) and at \(Q\) respectively make with the horizontal are equal.
Find an expression for the loss of kinetic energy when two imperfectly elastic spheres moving with given velocities impinge directly. An inelastic sphere of mass \(m\) is dropped with velocity \(V\) on the face of a smooth inclined plane of slope \(\alpha\) which is free to move on a smooth horizontal plane in a direction perpendicular to its edge. Shew that the loss of kinetic energy due to the impact is \[ \tfrac{1}{2}mMV^2\cos^2\alpha/(M+m\sin^2\alpha). \]
A particle is moving in a circle of radius \(r\) with velocity \(v\). Prove that its acceleration towards the centre is \(v^2/r\). A smooth circular tube is held fixed in a vertical plane. A particle of mass \(m\), which can slide inside the tube, is slightly displaced from rest at the highest point of the tube. Find the pressure between the particle and the tube when it is at an angular distance \(\theta\) from the highest point of the tube. Also find the vertical component of the acceleration of the particle when \(\theta=120^\circ\).