13. If, from the three angles of a triangle, lines be drawn to the points of bisection of the opposite sides, these lines intersect each other in the same point. 14. The three straight lines which bisect the three angles of a triangle, meet in the same point. 15. The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of two opposite sides, are, together, one half the parallelogram. 16. The figure formed by joining the points of bisection of the sides of a trapezium, is a parallelogram. 17. If squares be described on three sides of a rightangled triangle, and the extremities of the adjacent sides be joined, the triangles so formed are equivalent to the given triangle, and to each other. 18. If squares be described on the hypotenuse and sides of a right-angled triangle, and the extremities of the sides of the former, and the adjacent sides of the others, be joined, the sum of the squares of the lines joining them will be equal to five times the square of the hypotenuse. 19. The vertical angle of an oblique-angled triangle inscribed in a circle, is greater or less than a right angle, by the angle contained between the base and the diameter drawn from the extremity of the base. 20. If the base of any triangle be bisected by the diameter of its circumscribing circle, and, from the extremity of that diameter, a perpendicular be let fall upon the longer side, it will divide that side into segments, one of which will be equal to one half the sum, and the other to one half the difference, of the sides. 21. A straight line drawn from the vertex of an equilateral triangle inscribed in a circle, to any point in the opposite circumference, is equal to the sum of the two lines which are drawn from the extremities of the base to the same point. 22. The straight line bisecting any angle of a triangle inscribed in a given circle, cuts the circumference in a point which is equi-distant from the extremities of the side opposite to the bisected angle, and from the center of a circle inscribed in the triangle. 23. If, from the center of a circle, a line be drawn to any point in the chord of an arc, the square of that line, together with the rectangle contained by the segments of the chord, will be equal to the square described on the radius. 24. If two points be taken in the diameter of a circle, equidistant from the center, the sum of the squares of the two lines drawn from these points to any point in the circumference, will be always the same. 25. If, on the diameter of a semicircle, two equal circles be described, and in the space included by the three circumferences, a circle be inscribed, its diameter will be the diameter of either of the equal circles. 26. If a perpendicular be drawn from the vertical angle of any triangle to the base, the difference of the squares of the sides is equal to the difference of the squares of the segments of the base. 27. The square described on the side of an equilateral triangle, is equal to three times the square of the radius of the circumscribing circle. 28. The sum of the sides of an isosceles triangle is less than the sum of the sides of any other triangle on the same base and between the same parallels. 29. In any triangle, given one angle, a side adjacent to the given angle, and the difference of the other two sides, to construct the triangle. 30. In any triangle, given the base, the sum of the other two sides, and the angle opposite the base, to construct the triangle. 31. In any triangle, given the base, the angle opposite to the base, and the difference of the other two sides, to onstruct the triangle. TRIGONOMETRY. PART I. PLANE TRIGONOMETRY. SECTION I. ELEMENTARY PRINCIPLES. TRIGONOMETRY, in its literal and restricted sense, has for its object the measurement of triangles. When it treats of plane triangles it is called Plane Trigonometry. In a more enlarged sense, trigonometry is the science which investigates the relations of all possible arcs of the circumference of a circle to certain straight lines, termed trigonometrical lines or circular functions, connected with and dependent on such arcs, and the relations of these trigonometrical lines to each other. The measure of an angle is the arc of a circle intercepted between the two lines which form the angle-the center of the arc always being at the point where the two lines meet. The arc is measured by degrees, minutes, and seconds; there being 360 degrees to the whole circle, 60 minutes in one degree, and 60 seconds in one minute. Degrees, minutes, and seconds, are designated by °, ', "; thus, 27° 14′ 21′′, is read 27 degrees 14 minutes 21 seconds. The circumferences of all circles contain the same number of degrees, but the greater the radius the greater (244) is the absolute length of a degree. The circumference of a carriage wheel, the circumference of the earth, or the still greater and indefinite circumference of the heavens, has the same number of degrees; yet the same number of degrees in each and every circumference is the measure of precisely the same angle. DEFINITIONS. 1. The Complement of an arc is 90° minus the arc. 2. The Supplement of an arc is 180° minus the arc. 3. The Sine of an angle, or of an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end. Thus, BF is the sine of the arc AB, and also of the arc BDE. BK is the sine of the arc BD. 4. The Cosine of an arc is the perpendicular distance from the center of the circle to the sine of the arc; or, it is H G is the cosine of the arc AB; but CF= KB, which is the sine of BD. I 5. The Tangent of an arc is a line touching the circle in one extremity of the arc, and continued from thence, to meet a line drawn through the center and the other extremity. Thus, AH is the tangent to the arc AB, and DL is the tangent of the arc DB. 6. The Cotangent of an arc is the tangent of the complement of the arc. Thus, DL, which is the tangent of the arc DB, is the cotangent of the arc AB. REMARK. The co is but a contraction of the word complement. 7. The Secant of an arc is a line drawn from the center of the circle to the extremity of the tangent. Thus, CH is the secant of the arc AB, or of its supplement BDE. 8. The Cosecant of an arc is the secant of the complement. Thus, CL, the secant of BD, is the cosecant of AB. |