Calendar: 1874-1875 Page 552

552 GENERAL LITERATURE AND SCIENCE Show that log tan 10g tan0 2cosec2d 2cosec20 cot20 approximately when is small Under what circumstances does the principle of proportional parts fail to determine an angle fro η the logarithm of its tangent If be the angles of triangle "B nC cosn cos η Β cosn sin-ך- sin -5- sin -5 or nA nC cos cos cos -5- 'ύ 21 21 according as η is odd or even Show how to express the cosine an angle of triangle in terms of the sides If Ρ be point within triangle whose distances from the angular points are β and from the centre of the circ in- scribing circle show that a2 0P Λ2 si112 0י ϋη sin2B y2 rf2 E1 sin2 where is the radius of the circumscribing circle person on the slope of hill observes the angles of eleva- tion and β of two objects on the hill and also the angle γ which they subtend at his position if θ be the inclination of the hill to the horizon show that 8in20 sinsy sin -f- sin2 2sina sin cosy Prove the exponential expressions for sin and costf and apply them to show that x3 rcos j9 p5 cos 2β cos 3j3 etc to infinity cos Cos -f- xsmj3 cosa Through the angular points of triangle lines are drawn perpendicular to the lines which bisect the angles respectively and produced to meet so as to torm another triangle if Δ' be the area of the new triangle and Δ that of the original one show that י ABC Δ Δ coseo pose cosep י it