Sum the following series:
Prove that \(\displaystyle\frac{\sin n\theta}{\sin\theta}\) is divisible by \(\cos\theta-\cos\alpha\), where \(n\) is an integer, and shew that \[ \frac{\sin n\theta}{\sin\theta} = 2^{n-1}\left(\cos\theta-\cos\frac{\pi}{n}\right)\left(\cos\theta-\cos\frac{2\pi}{n}\right)\dots\left(\cos\theta-\cos\frac{n-1}{n}\pi\right). \] Prove that \[ 32\cos\frac{\pi}{11}\cos\frac{2\pi}{11}\cos\frac{3\pi}{11}\cos\frac{4\pi}{11}\cos\frac{5\pi}{11}=1. \]
The diagonals of a parallelogram are the straight lines whose equation referred to rectangular coordinates is \[ ax^2+2hxy+by^2=0, \] and \((\alpha,\beta)\) are the coordinates of the middle point of one side of the parallelogram. Find the equation of that side and shew that one of the angles of the parallelogram is \[ \tan^{-1}\frac{a\alpha^2+2h\alpha\beta+b\beta^2}{h(\alpha^2-\beta^2)+\alpha\beta(b-a)}. \]
Find the equations of the tangent and normal at the point \((h,k)\) on the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \] If the tangent at \(P\) cuts the directrices in \(Q\) and \(Q'\), prove that the other tangents from \(Q\) and \(Q'\) to the ellipse meet on the normal at \(P\).
Find the coordinates of the focus of the parabola \[ (x\sin\theta+y\cos\theta)^2=4ay\sin\theta, \] referred to rectangular axes; and shew that the equation of the latus rectum is \[ x\cos\theta-y\sin\theta=a\cos2\theta. \]
Find the conditions that the equation \[ Ax^2+By^2+Cz^2+2Fyz+2Gzx+2Hxy=0, \] in areal coordinates, represents a circle. Prove that, if \(ABC\) is the triangle of reference, \[ (y\cot B-z\cot C)^2=x(y+z) \] is the equation of the circle described on the perpendicular from \(A\) on the side \(BC\), as diameter; and find the coordinates of the point of intersection of the tangents to this circle at the points at which it cuts the sides \(AB\) and \(AC\).
Prove that, if \[ y=(\sin^{-1}x)^2, \] then \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 2, \] and \[ (1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+1)x\frac{d^{n+1}y}{dx^{n+1}} - n^2\frac{d^ny}{dx^n}=0. \]
The two chains of a suspension bridge hang in a parabola of span 80' and dip 16'; they are stiffened by a roadway, rigid except for hinges at the centre and at each end. If the chains and roadway weigh 0.3 ton per foot of span, find the maximum tension in each chain when a 2 ton concentrated load crosses the bridge, and the maximum bending moment on the roadway.
The countershaft of a lathe carries two gear-wheels whose pitch diameters are 6" and 3" respectively: the central planes of the wheels are 6" apart and the ends of the shafts run in bearings outside the wheels whose central planes are 2" from the central plane of the corresponding wheel. The gear-wheels on the mandrel have pitch diameters of 3" and 6", so that the total reduction ratio is 4:1, and the driving belt exerts a couple of magnitude 20 lb. ft. on the first of these two latter wheels. If the angle of obliquity of thrust between the gear teeth be 20\(^\circ\), calculate the load on each bearing of the countershaft, neglecting friction throughout.
State and prove any graphical construction for finding the acceleration of a steam-engine piston at any point of the stroke. Find the acceleration of the middle point of the connecting rod of an engine running at 400 r.p.m. at the instant when the crank is at right angles to the line of stroke, if the lengths of crank and connecting rod be 6" and 2' 0" respectively.