Find the fourth differential coefficient of \(\frac{\sin x}{x}\); and deduce that as \(x\to 0\), \[ \frac{x^4-12x^2+24}{x^5}\sin x + \frac{4x^2-24}{x^4}\cos x \to -\frac{1}{5}. \]
A triangle is circumscribed to a circle of given radius \(r\), and the sides of the triangle are to be determined in terms of \(r\) and the angles by the formula \[ r = a(\cot\frac{1}{2}B + \cot\frac{1}{2}C), \] and others like it. A first measurement makes the triangle equilateral. Shew that, if there is a possible error of \(10'\) in each of the angles \(B\) and \(C\), the percentage of error in the determination of the side \(a\) cannot exceed \(\cdot34\).
Evaluate \(\int\sec^3 x dx, \int\frac{3x+2}{\sqrt{\{x^2+4x+1\}}}dx\). Prove that \[ \int_1^\infty \frac{x^2+2}{x^4(x^2+1)}dx = \frac{\pi}{4}-\frac{1}{3}. \]
Prove that the area of the curved surface and the volume of a segment of height \(h\) of a sphere of radius \(a\) are \(2\pi ah\) and \(\frac{1}{3}\pi h^2(3a-h)\). The whole area (curved and plane) of a segment of a sphere is given to be equal to \(\pi c^2\). Prove that when the volume is greatest the height of the segment is \(c\).
Trace the curve \(x^3+y^3-2ax^2=0\).
Find an expression for all the values of \(\theta\) satisfying the equation \(\sin\theta=\sin\alpha\). If \(\theta_1, \theta_2\) are the two values of \(\theta\) not differing by a multiple of \(\pi\) which satisfy the equation \[ \frac{\sin\theta\sin\phi}{\sin\alpha} + \frac{\cos\theta\cos\phi}{\cos\alpha} + 1 = 0, \] prove that \[ \cos(\theta_1+\theta_2) = \frac{\sin^2\alpha-\sin^2\phi}{\sin^2\alpha\cos^2\phi+\cos^2\alpha\sin^2\phi}. \]
In any triangle prove the formulae
Prove that \((\cos\theta+i\sin\theta)^{p/q}\), where \(p\) and \(q\) are integers, has \(q\) values. Find all the values of \((\sqrt{-1})^{\sqrt{-1}}\).
\(S\) is the area of a quadrilateral of which \(a,b,c,d\) are the sides, \(x,y\) the diagonals, and \(2\alpha\) the sum of two opposite angles, prove that \begin{align*} S^2 &= (s-a)(s-b)(s-c)(s-d)-abcd\cos^2\alpha, \\ x^2y^2 &= (ac+bd)^2 - 4abcd\cos^2\alpha, \end{align*} where \(2S = a+b+c+d\).
Prove that the sum of the moments of a system of two intersecting forces about any point in their plane is equal to the moment of their resultant. Forces \(P, 2P, 3P, 4P\) act along the sides \(AB, BC, CD, DA\) respectively of the square \(ABCD\). Find the magnitude of their resultant and the points in which it cuts the sides \(AB\) and \(BC\).