\(ABC\) is a triangle, and \begin{align*} \sin A + \sin B + \sin C &= p, \\ \cos A + \cos B + \cos C &= q. \end{align*} Express \[ \sin 2A + \sin 2B + \sin 2C \quad \text{and} \quad \cos 2A + \cos 2B + \cos 2C \] in terms of \(p\) and \(q\).
Solution: Consider \(q+ip = e^{iA}+e^{iB}+e^{iC}\), we would like \begin{align*} e^{i2A} + e^{i2B}+e^{i2C} &= ( e^{iA}+e^{iB}+e^{iC})^2 -2(e^{i(A+B)}+e^{i(B+C)}+e^{i(C+A)}) \\ &= (q+ip)^2-2(e^{i(\pi-C)}+e^{i(\pi-A)} + e^{i(\pi-B)})\\ &= (q+ip)^2+2(e^{-iC}+e^{-iA}+e^{-iB}) \\ &= (q+ip)^2+2(q-ip) \\ &= q^2-p^2+2q+i(2pq-p) \end{align*}
Express \(\log_e(a+b\sqrt{-1})\) in the form \(x+y\sqrt{-1}\). \par Find the value of \(\log_e(-1)\), and point out the fallacy in the following: \[ \log_e(-1) = \tfrac{1}{2}\log_e(-1)^2 = \tfrac{1}{2}\log_e 1=0, \] \[ \therefore -1 = e^0 = 1. \]
Prove that three normals can be drawn to a parabola from a given point. \par The normals at \(P, Q, R\) to a parabola whose vertex is \(A\) intersect in \(T\). A second parabola is drawn through \(P, Q, R\) with its axis perpendicular to the axis of the first parabola. Prove that the second parabola passes through \(A\) and that the length of its latus rectum is equal to half the distance of \(T\) from the axis of the first parabola.
Find the equation of the tangent at any point of the curve \(y^2=x^3\). \par The tangent at \(P\) intersects the curve again at \(Q\), and the lines from the origin to \(P\) and \(Q\) make angles \(\alpha, \beta\) respectively with the axis of \(x\). Shew that \[ \tan\alpha+2\tan\beta=0. \]
A manufacturer's expenses are a fixed sum together with a fixed amount \(c\) for each article sold. The number of articles sold varies as the \(k\)th power of the amount by which the selling price of an article is less than a fixed sum \(p\). Shew that his profits will be greatest if the selling price of an article is \(\frac{p+ck}{1+k}\).
Four forces act at the middle points of the sides of a quadrilateral figure in directions at right angles to the respective sides. Prove that if the forces are proportional to the respective sides they will be in equilibrium. \par Prove also that if the forces are in equilibrium they must be proportional to the sides unless the quadrilateral is such that a circle can be described about it.
A heavy beam inclined at an angle \(\alpha\) to the horizontal rests with one end against a vertical wall and the other on the ground, the coefficient of friction in each case being \(\tan\lambda\). The beam is in a vertical plane perpendicular to the plane of the wall. Shew that if the beam is kept from slipping down by a horizontal string of tension \(T\) attached to the lower end, or by a vertical string of tension \(T'\) attached to the upper end, then \[ T'=T\tan(\alpha+\lambda). \]
When a ship is steaming due North the line of smoke makes an angle \(\alpha\) to the East of South; on the ship turning due East the line of smoke makes an angle \(\beta\) to the South of West; on its turning due South the smoke makes an angle \(\gamma\) to the East of North. Shew that \[ \cot\beta(\cot\alpha-\cot\gamma) = \cot\alpha+\cot\gamma-2. \]
Solve the equations:
Find the sum of the squares of the first \(n\) odd numbers. \par Prove that the sum of the squares of all numbers less than 60 and prime to 60 is 19120.