Prove that, if \[ \sin(x+\alpha) + \sin(x+\beta) + \sin(x+\gamma) + \sin(x+\delta) = 0, \] and \[ \cos(x+\alpha) + \cos(x+\beta) + \cos(x+\gamma) + \cos(x+\delta) = 0, \] then \(\pm\alpha\pm\beta\pm\gamma\pm\delta = 2n\pi\), where two of the signs are plus and two are minus. Solve the equation \[ (\cos x + \cos 3x)(\cos 2x + \cos 4x) = \frac{1}{4}. \]
Find expressions for the sides and angles of the pedal triangle of a triangle ABC. Shew that, if O is the circumcentre and R the circumradius of ABC and if H is the incentre and \(\rho\) the inradius of the pedal triangle, then \[ R^2 = OH^2+4R\rho. \]
Find the coefficient of \(x^n\) in the expansions in ascending powers of \(x\) of each of the following functions: \[ \text{(i)} \quad \frac{1-x^2}{1-2x\cos\theta+x^2}, \quad \text{(ii)} \quad \log_e(1+2x\cos\theta+x^2), \quad \text{(iii)} \quad e^{x\cos\alpha}\sin(\alpha+x\sin\beta). \]
To an observer walking along a straight level road PQR three mountain peaks A, B, C are visible in a plane containing the road. At P the peak B just conceals A, at Q the peak C just conceals A and at R the peak C just conceals B. Prove that, if f, g, h are the heights of the peaks A, B, C they are connected by the relation \[ f(g-h):h(f-g) = PQ:QR. \]
ABCD is a rhombus of smoothly jointed rods resting on a smooth horizontal table to which CD is fixed. The points A, C are joined by an inelastic string which is kept taut by a force F applied at right angles to AB at its middle point. Shew that the forces along the rods BC, AD are in the ratio \(\sin\alpha : \sin 3\alpha\), where the angle CAB = \(\alpha\).
Explain what is meant by the angle of friction. A uniform plank of length \(l\) and thickness \(2h\) rests symmetrically across a fixed rough cylinder of radius \(a\). Taking \(\lambda\) to be the angle of friction between the bodies, find the relation between \(\lambda, a\) and \(h\) in order that if the plank be slowly tilted another position of equilibrium may be reached, and shew that if \(a\) were less than \(h\) no amount of friction would make this possible.
What is the energy test of stability of equilibrium? How is it connected with the principle of conservation of energy? A uniform solid hemisphere hangs with its plane face towards a smooth vertical wall suspended by a string of length equal to the diameter, one end being fastened to the wall and the other to the rim of the hemisphere. Find the inclination of the string to the wall in equilibrium and determine whether the equilibrium is stable or not.
Define the angular velocity of a body moving in any manner in a plane. A circular ring of radius \(b\) turns round a fixed point O in its circumference with uniform angular velocity \(\Omega\). A smaller ring of radius \(a\) rolls on the inside of the larger ring with uniform angular velocity \(\omega\), the angular velocities being in the same sense. Find the velocity of any point of the smaller ring in any position. Also shew that, if \(a\omega = b\Omega\), then in every position of the smaller ring one point on it is at rest. Indicate the position of this point for a general position of the rings.
A train of 200 tons, uniformly accelerated, acquires in two minutes from rest a velocity of 30 m.p.h. Shew that, if the coefficient of friction be \(\cdot 18\), the part of the load carried by the driving wheels of the engine cannot be less than 12.7 tons.
A body is projected from the ground with velocity \(u\) at inclination \(\alpha\) to the horizontal. At the highest point of the trajectory the body is broken into two parts by an internal explosion which creates E foot-pounds of energy without altering the direction of motion. Shew that the distance between the parts when they reach the ground is \[ 2\left(\frac{E}{mg}\right)^{\frac{1}{2}} u \sin\alpha, \] where \(m\) is the harmonic mean of the masses of the parts.