A Collection of Examples in Pure and Mixed Mathematics, with Hints and Answers, by A. Wrigley and W. H. Johnstone

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1845 edition. Excerpt: ...powers of sin 9 and cos 0. 242. Find tan 3 0, and tan 8 0 in terms of the powers of tan 0. From the exponential expressions for sin 0 and cos 0, ob-tain the following formulae: --2tan0 ANALYTICAL GEOMETRY OF TWO DIMENSIONS. APPLICATION OF ALGEBRA TO GEOMETRY. 1. To describe a square in a given triangle. 2. Given the hypothenuse of a right-angled triangle, and the side of an inscribed square: required the two sides of the triangle, (1) when the given side coincides with the hypothenuse, and (2) when it does not so coincide. 3. In a given square to inscribe another square having its side equal to a given straight line. To what limitation is this line subject? 4. To find a triangle such that its sides and a perpendicular on one of them from the opposite angle are in continued geometrical progression. 5. Through a point M equidistant from two straight lines AA', BB' at right angles to each other, to draw a straight line PMQ, so that the sum of the squares upon PM and MQ shall be equal to the square upon a given line b. 6. To inscribe a semicircle in a quadrant. 7. To inscribe a circle in a given sector of a circle. 8. If a, b, c be the chords of three adjacent arcs of a circle whose sum equals the semicircumference, of which x is the radius, then will 4r'--(a1 + b1 + c1)x--abc= 0. 9. Given the three perpendiculars from the angles of a triangle upon the opposite sides; to find the area and the sides of the triangle. 10. Given the chords of two arcs of a given circle; to find the chord of their sum, and the chord of their difference. 11. Having given the lengths of two chords which intersect at right angles, and the distance of their point of intersection from the centre; to find the diameter of the circle. 12. If r be the radius of a circle...