Prove that the sine, cosine and tangent of any multiple of \(\theta\) are rational algebraic functions (i) of \(\tan\frac{1}{2}\theta\), (ii) of \(\cos\theta + i\sin\theta\). In what way are these facts applied to the solution of trigonometrical equations and to the integration of trigonometrical functions? Shew that the equation \[ \cos(2\theta - \alpha) = k \cos(\theta - \beta) \] can be satisfied by four values \(\theta_1, \theta_2, \theta_3, \theta_4\) of \(\theta\), of which no two differ by a multiple of \(2\pi\), and by no more. Shew also that \[ \theta_1 + \theta_2 + \theta_3 + \theta_4 - 2\alpha = 2n\pi, \] and that \[ \cos(\theta_2 + \theta_3 - \alpha) + \cos(\theta_3 + \theta_1 - \alpha) + \cos(\theta_1 + \theta_2 - \alpha) = 0. \]
A circular hill is very nearly of the form given by a regular truncated cone 3000 ft. in diameter at the base and 1000 ft. in diameter at the top, the height being 40 ft. A railway cutting is made through the hill, the bottom of the cutting being level with the base and 25 ft. wide, whilst the width on the flat top of the hill is 100 ft. Estimate the volume of earth to be excavated.
Four points \(A, B, C, D\) of a conic have the property that, if \(P\) is any point on the curve, \(PA\) and \(PB\) are harmonically conjugate to \(PC, PD\). By considering special positions of \(P\), shew that the tangents at \(A\) and \(B\) intersect upon \(CD\). Hence or otherwise shew how to find by ruler and compass those points of a given conic at which (i) two coterminous arcs of the curve \(AC, CB\), (ii) any two arcs of the curve, \(A'A', B'B\), subtend equal angles.
Two rough planes are equally inclined at an angle \(\alpha\) to the horizontal. A cylinder of radius \(a\), whose centre of mass is at distance \(c\) from its axis, rests between them. If \(\lambda\) be the angle of friction between the cylinder and each plane \((\lambda < \frac{1}{2}\pi - \alpha)\), shew that it is impossible for the cylinder to be placed in any position of limiting equilibrium if \[ c < a \sin \alpha \sin 2\lambda / \sin 2\alpha. \]
Prove that if \(a, b, c\) are in arithmetical progression, and \(a, b, d\) in harmonical progression, then \[ \frac{c}{d} = 1 - \frac{2(a-b)^2}{ab}. \]
Consider some of the chief results and formulae of analytical geometry in rectangular cartesian coordinates (concerning for example parallel, perpendicular or concurrent lines; the centres, foci, tangents, normals, conjugate lines and points of conics; lengths, areas, and their centres of gravity etc.) and, by drawing up two short lists, distinguish those where the results hold without modification for oblique cartesian coordinates from those where the results do not so hold. What geometrical process can be applied to the former class without destroying their properties and not to the latter? Illustrate the main differences between the two classes.
A string hung from two fixed points in the same horizontal line carries weights of 3, 2, 5, 2, 3 lbs. arranged in this order. If the inclination to the horizontal of each of the two portions of the string which support the 5 lbs. mass be 10\(^{\circ}\), find the inclination of the two end portions of the string which are attached to the fixed points, and the tensions in the string at those points.
Eliminate \(x, y, z\) from the equations \begin{align*} (z + x - y) (x + y - z) &= ayz, \\ (x + y - z) (y + z - x) &= bzx, \\ (y + z - x) (z + x - y) &= cxy, \end{align*} shewing that the result is \[ abc = (a + b + c - 4)^2. \]
The diagram represents a framework of smoothly jointed rods, loaded at \(CDE\), and supported at \(A\) and \(B\). If \(\alpha=60^{\circ}\), draw a diagram giving the stresses in the rods. Shew that if \(\tan\alpha=3\) there is no stress in the rod \(FD\). [Diagram of a truss with supports at A and B. From A and B, members go up to C and E respectively at 45 degrees. C, D, E are connected horizontally. D is the midpoint of CE. From C, D, E members go up to a peak F, forming triangles. The angle at F is \(\alpha\). Loads W act downwards at C, D, and E.]
Explain the principle of proof by 'mathematical induction'; and prove in this way that \[ 1-\frac{1}{2}+\frac{1}{3}-\dots-\frac{1}{n} = 2\left(\frac{1}{n+2} + \frac{1}{n+4} + \dots + \frac{1}{2n}\right) \] if \(n\) is even.