Eliminate \(\theta\) from the equations \[ a\sin\theta + b\cos\theta = a\operatorname{cosec}\theta - b\sec\theta = 1. \] Prove that \[ \tan(\alpha-\beta)+\tan(\beta-\gamma)+\tan(\gamma-\delta)+\tan(\delta-\alpha) + \frac{\sin(\alpha-\beta+\gamma-\delta)\sin(\alpha-\gamma)\sin(\beta-\delta)}{\cos(\alpha-\beta)\cos(\beta-\gamma)\cos(\gamma-\delta)\cos(\delta-\alpha)} = 0. \]
Prove that the length of the line joining the orthocentre of a triangle \(ABC\) to the middle point of the side \(BC\) is \[ \frac{a}{2}\left\{\frac{1-4\cos A \cos B \cos C}{\sin^2 A}\right\}^{\frac{1}{2}}. \] Prove that, if the side \(a\) of a triangle \(ABC\) is increased by a small quantity \(x\) while the other two sides remain constant, the radius of the circumscribing circle will be increased by approximately \[ \frac{1}{2}x \operatorname{cosec} A \cot B \cot C. \]
Express \(\cos 7\theta\) in terms of \(\cos\theta\). Shew that \(\cos\frac{\pi}{7}\) is a root of the equation \[ 8x^3 - 4x^2 - 4x + 1 = 0, \] and state what are the other roots of this equation.
Forces \(P, Q, R\) act along the sides \(BC, CA, AB\) of a triangle \(ABC\), and forces \(P', Q', R'\) act along \(OA, OB, OC\) where \(O\) is the centre of the circumscribing circle. Prove that, if the six forces are in equilibrium, \[ P\cos A + Q\cos B + R\cos C = 0, \] \[ \frac{PP'}{a} + \frac{QQ'}{b} + \frac{RR'}{c} = 0. \] Distinguish between sufficient and necessary conditions of equilibrium, and illustrate from this question.
A framework of four heavy rods, of length \(a\), hinged together to form a rhombus is supported by a smooth cylinder of radius \(c\). Shew that, if the rods are in equilibrium when each makes an angle 30\(^{\circ}\) with the vertical, then \(a=4\sqrt{3}c\). Find the ratio of the actions at the top and bottom hinges.
Determine the conditions of equilibrium for a system of forces not in one plane. A heavy sphere rests on three pegs \(A, B, C\) in a horizontal plane. Prove that the pressures on the pegs are proportional to \(\sin 2A, \sin 2B, \sin 2C\). Shew what can be proved about the pressures if the sphere rests on four pegs in the same horizontal plane.
A mass \(M\) is fastened to one end of a fine string which passes over a smooth pulley, and to the other end of the string is attached a smooth pulley: over this second pulley a fine string passes, one end of which is fastened to the ground and the other end to a mass \(m\). Determine the motion, and shew that if \(M=4m\) the tension of the string passing over the fixed pulley is \(3mg\).
Find the horse-power of an engine which can just pull a train of \(m\) tons with velocity \(v\) miles per hour up an incline of 1 in \(n\); the resistance to motion being \(x\) lbs. per ton on the level. If \(v=40\) miles per hour when \(x=15\) and \(n=500\), shew that the engine will be able to pull the train down the same incline at 74 miles per hour approximately.
A particle is projected from any point of an inclined plane in a direction in the same vertical plane as the line of greatest slope through the point: find the range up the plane. The sighting of a gun is correct when it is used for firing at objects in the same horizontal plane, and when the range is 6000 yards the initial direction of a shot is inclined at 30\(^{\circ}\) to the horizontal. Shew that, if the gun is used to fire at a point 6000 yards away and 120 yards higher than the firing point, a man aiming at such a point must have the gun sighted for 6070 yards approximately.
Prove that when a body describes a path round a centre of force the radius vector of the path sweeps out equal areas in equal times. Taking the semi-axis major of the earth's orbit round the sun as 93,000,000 miles and the eccentricity as \(\frac{1}{60}\), find in miles per second the difference between the velocities at the two ends of the major axis.