Prove that the centre of the circle inscribed in the triangle formed by the external common tangents to the escribed circles of any triangle \(ABC\) is the point of intersection of the perpendiculars from the centres of the escribed circles upon the corresponding sides, and that the radius of this circle is equal to \(R(1+\cos A + \cos B + \cos C)\), where \(R\) is the radius of the circle circumscribing the triangle \(ABC\).
Any number of forces \(P_1, P_2, \dots, P_n\) in the same plane are in equilibrium. The direction of each is given, and also the magnitude of \(P_1\) and the ratios of the magnitudes of \(P_3, P_4, \dots, P_n\). Shew how to construct the magnitude of each force.
Three equal spheres rest in contact on a rough horizontal plane. An equal sphere of the same material is placed so as to rest symmetrically on them. Shew that, if the coefficient of friction \(\mu\) is \(>\sqrt{3}-\sqrt{2}\) and all surfaces are equally rough, equilibrium will be maintained.
Two equal particles \(A, B\) attached to the ends of a light string of length \(a\) are placed on a smooth horizontal table with the string \(AB\) perpendicular to the edge of the table and \(B\) hanging just over the edge. The system is released from rest in this position. Prove that when first the string is horizontal the distance of \(B\) from the vertical through the edge of the table is \(\frac{1}{4}a(\pi-2)\), and find the tension in the string.
A heavy particle is supported in equilibrium by two equal elastic strings with their other ends attached to two points in a horizontal plane and each inclined at an angle of \(60^\circ\) to the vertical. The modulus of elasticity is such that when the particle is suspended from any portion of the string its extension is equal to its natural length. The particle is displaced vertically a small distance and then released. Prove that the period of its small oscillations is \(2\pi\sqrt{2l/5g}\), where \(l\) is the stretched length of either string in equilibrium.
A particle of mass 2 lb. is placed on the smooth face of an inclined plane of mass 7 lb. and slope \(30^\circ\), which is free to slide on a smooth horizontal plane in a direction perpendicular to its edge. Shew that if the system start from rest the particle will slide down a distance of 15 feet along the face of the plane in 1.25 seconds.
Find the values of \(\sin 15^\circ\) and \(\sin 18^\circ\). If \[ \cos(\theta-\phi)/\cos(\theta+\phi) = a/b, \] prove that \[ (a^2+b^2-2ab\cos 2\theta)(a^2+b^2-2ab\cos 2\phi) = (a^2-b^2)^2. \]
In any triangle prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}, \quad s=4R\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}. \] A perpendicular \(CD\) is drawn from the vertex \(C\) of a right-angled triangle on the hypotenuse \(AB\). Prove that the square of the radius of the circle inscribed in \(ABC\) is equal to the sum of the squares of the radii of those inscribed in \(ACD\) and \(BCD\).
Expand \(\cos n\theta\) in a series of ascending powers of \(\cos\theta\). Prove that \[ \sum_{r=0}^{r=n-1} \sec^2\left(\theta+\frac{2r\pi}{n}\right) = n^2\sec^2 n\theta, \] when \(n\) is odd, and find its value when \(n\) is even.
Prove that \((\cos\theta+i\sin\theta)^{p/q}\) has \(q\) values, where \(p,q\) are integers and \(q\) is prime to \(p\). If \[ \tan^{-1}(\xi+i\eta) = \sin^{-1}(x+iy), \] prove that \[ \xi^2+\eta^2 = (x^2+y^2)/\sqrt{x^4+y^4+2x^2y^2-2x^2+2y^2+1}. \]