Find the first significant term in the expansion in ascending powers of \(\theta\) of \[ \frac{2\theta - 28\sin\theta+\sin 2\theta}{9+6\cos\theta}. \]
The radii of two parallel plane sections of a sphere are \(a,b\), and the distance between them is \(c\). Prove that the included volume is \(\frac{1}{6}\pi c(c^2+3a^2+3b^2)\), and the included surface \(\pi\{(a^2+b^2+c^2)^2-4a^2b^2\}^{\frac{1}{2}}\). What is the condition that the centre of the sphere should lie within the volume considered?
Prove that if \(\cos 2A + \cos 2B + \cos 2C + 4\cos A \cos B \cos C + 1 = 0\) then \(A \pm B \pm C\) must be an odd multiple of 180\(^\circ\). Prove that \begin{align*} \Sigma \sin^2 A \sin(B+C-A) &- 2\sin A \sin B \sin C \\ &= \sin(B+C-A)\sin(C+A-B)\sin(A+B-C). \end{align*}
Shew how to solve a triangle \(ABC\) having given \(B-C, b-c\) and the perpendicular distance of \(A\) from \(BC\), and adapt the solution to logarithmic computation.
Prove that, if \(x<1\) \[ \frac{1-x^2}{1-2x\cos\theta+x^2} = 1+2x\cos\theta+2x^2\cos 2\theta+2x^3\cos 3\theta+\dots. \] Obtain the first 3 terms of the expansion of \(\cos n\theta\) in ascending powers of \(\cos\theta\), when \(n\) is even.
Find the real quadratic factors of \(x^{2n} - 2x^n\cos n\alpha + 1\). Prove that, if \(n\) is an odd integer \[ \sin^2\theta \sin^2(\theta+\frac{\pi}{n})\dots\sin^2(\theta+\frac{n-1}{n}\pi) + \cos^2\theta\cos^2(\theta+\frac{\pi}{n})\dots\cos^2(\theta+\frac{n-1}{n}\pi) = 2^{2-2n}. \]
Shew that the effect of a couple is independent of its position in the plane in which it acts. \(ABCD\) is a skew quadrilateral. Prove that forces completely represented by the lines \(AB, BC, CD, DA\) are equivalent to a couple in a plane parallel to \(AC\) and \(BD\) and determine its moment.
A drawer of depth \(b\) (from back to front) is jammed by pulling at a handle at a distance \(c\) from the centre of the front. Prove that the coefficient of friction must be at least \(b/2c\).
Prove that when any system of bodies is suspended under the action of gravity and their mutual reactions, the position in which the depth of the centre of gravity below a fixed level is a maximum is a position of equilibrium. A chain of five uniform equal rods smoothly jointed is hung from its ends at two points in the same horizontal line. Use the foregoing principle to shew that the inclinations of the rods to the horizontal are connected by a relation \(\tan\alpha = 2\tan\beta\).
Find the Horse Power of an engine required to pump out a dock 300 feet long, 90 feet wide and 20 feet deep in 3 hours, if the water is delivered through a pipe of section one square foot at a level 30 feet above the bottom of the dock, the efficiency of the pump being 75 per cent. and account being taken of the energy of the water as it leaves the pipe.