same circle, what will be the number of sides, when the whole space intercepted between the two polygonal boundaries is an assigned part of either polygon? Ex. What is the figure when the exterior area is of the interior polygon? (6) In a right-angled triangle, if the hypothenuse (c) be divided into segments (x), (y) by the line which bisects the right angle, and t the tangent of half the difference of the acute angles, x y1t: 1-t. (7) To divide a given angle into two others, whose sincs shall be in the ratio of m: n. (8) A circle is inscribed in an equilateral triangle, an equilateral triangle in the circle, a circle again in the latter. triangle, and so on; if r, 71, 72, 73, &c., be the radii of the circles, shew that r = r1 + r2+ry + &c. (9) If a, b, c be the sides of a triangle, and p, q, r perpendiculars from a point within the triangle bisecting the sides, prove that (10) It is required to determine the continued product of n terms of the series 1 1 (sin cos) (sin cos) (sin cos 1 1 4 (11) The circumference of the inner of two concentric circles, of which R and r are the radii, is divided into n equal parts; shew that the sum of the squares of the lines drawn from the points of division to any point in the circumference of the outer circle = n (R2 + r2). (12) The ratio between the area of an equilateral and equiangular decagon described about a circle, and that of another within the same circle, is equal to (13) In a plane triangle ABC, having given the sum of the sides AC, CB, the perpendicular from the vertex C upon the base AB; and the difference of the segments of the base made by the perpendicular; find the sides of the triangle. (14) The length of a road in which the ascent is 1 foot in 5, from the foot of a hill to the top is a mile and twothirds. What will be the length of a zigzag road, in which the ascent is 1 foot in 12? (15) At each end of a horizontal base measured in a known direction from the place of an observer, the angle which the distance of the other end and a certain object subtends is observed, as also the angle of elevation of the object at one end of the base. Find its height and bearing. (16) A staff one foot long stands on the top of a tower 200 feet high. Shew that the angle which it subtends at a point in the horizontal plane 100 feet from the base of the tower is nearly 6'51". (17) Find the degrees, minutes, and seconds, in the angle whose circular measure is 1; also the values of cos mπ and tan-(-1)m (m an integer). (18) Two circles have a common radius (r) and a circle is described touching this radius and the two circles: prove that the radius of the circle which touches the three = 3r 56* (19) At noon, a column in the E.S.E. cast upon the ground a shadow, the extremity of which was in the direction N.E.: the angle of elevation of the column being a°, and the distance of the extremity of the shadow from the column (a) feet, determine the height of the column. (21) If A, B, C be the angles of a triangle, (22) In a regular polygon of n sides inscribed in a circle. whose radius is (r), if a be the distance of any point from the centre, and perpendiculars be drawn from this point upon all the sides of the polygon, the sum of the squares of the lines joining the feet of the adjacent perpendiculars is (23) Four objects situated at unequal but given distances in the same straight line, appear to a spectator in the same plane with them to be at equal distances from each other, it is required to determine his position. (24) If a', b' be the segments of the hypothenuse made by a line bisecting the right angle, then (25) A boy flying a kite at noon, when the wind was blowing a from the south, and the angular distance of the kite's shadow from the north was ẞ, the wind suddenly changed to a from the south, and the shadow to ẞ, from the north, and the kite was raised as much above 45° as it had before been below that elevation. Shew that 0° being the angular elevation of the sun, and 45° — go that of the kite at first, INDEX. Abbreviation of trigonometrical quantities. 8. Angle, circular measure of the, 3; divi- Angles, relation between the angles and Arc, series for sine and cosine in terms Area of polygons, to find the, 80. biguous case, 58; to find the area of a Centisimal division, 1. Cosecart, definition of, 8; ratio of, 8; Cosine of 150, 180, 360, 540, 749, Cosine and sine, series for, in the terms of the arc, 109; remarks on the object Cosine and sine of A B, to find the ex- Cosine, to find the, and sine, of the sum 28. Cosine, to find, of an angle of a plane Definitions, 8. Demoivre's theorem explained, 104. Distances, trigonometrical, calculation of: English angular measure, described, 1; Examples, miscellaneous, 116. Formulæ expressing the sine, cosine, and Formulæ of verification, 33. French angular measures described, 1; Heights, trigonometrical, calculation of: Inverse trigonometrical functions de- Miscellaneous examples, 116. Measure, circular, of the angle, 3. |