VIII. Having given two sides and the included angle of a triangle, to describe the triangle. 55. Let the line B=150 feet, and 120 feet, be the given sides; and A=30 degrees, the given angle: to describe the triangle on a scale of 200 feet to the inch. Draw the indefinite line DG, and at the point D, make the angle GDH equal to 30 degrees: then lay off DG equal to 150, equal to three quarters of an inch, and DH equal to 120, equal to six tenths of an B C H inch, and draw GH: DHG will be the required triangle. IX. The three sides of a triangle being given, to describe the triangle. 56. Let A, B and C, be the sides. D AH E Draw DE equal to the side A. From the point D as a centre, with a radius equal to the second side B, describe an arc: from E as a centre, with a radius equal to the third side C, describe another arc intersecting the former in F; draw DF and EF, and DFE will be the triangle required. BH CH X. Having given two sides of a triangle and an angle opposite one of them, to describe the triangle. 57. Let A and B be the given sides, and the given angle, which we will suppose is opposite the side B. Draw the indefinite line DF and make the angle FDH equal to the angle C: take DH= A, from the point H, as a centre, with a radius equal to the other given side, B, describe an arc cutting A BH H D F DF in F; draw HF: then will DHF be the required tri If the angle C is acute, and the side B less than A, then the arc described from the centre E with the radius EF =B will cut the side DF in two points, F and G, lying on the same side of D: AH B D hence, there will be two triangles, DEF, and DEG, either of which will satisfy all the conditions of the problem. XI. The adjacent sides of a parallelogram, with the angle which they contain, being given, to describe the paral lelogram. 58. Let A and B be the given sides, and a the given angle. Draw the line DH, and lay off DE equal to A; at the point D, make the angle EDF= 0; take DFB: describe two arcs, the one from D AH BH F E H F, as a centre, with a radius FG = DE, the other from E, as a centre, with a radius EG = DF; through the point G, where these arcs intersect each other, draw FG, EG; DEGF will be the parallelogram required. XII. To find the centre of a given circle or arc. 59. Take three points, A, B, C, any where in the circumference, or in the arc: draw AB, BC; bisect these two lines by the perpendiculars, DE, FG: the point 0, where these perpendiculars meet, will be the centre sought. The same construction serves for making a circumference pass through three given points A, B, C, and also for describing a circumference, about a given PLANE TRIGONOMETRY. SECTION III. DEFINITIONS.—APPLICATION TO HEIGHTS AND DISTANCES. 1. In every plane triangle there are six parts: three sides and three angles. These parts are so related to each other, that when one side and any two other parts are given, the remaining ones can be obtained, either by geometrical construction or by trigonometrical computation. 2. Plane Trigonometry explains the methods of computing the unknown parts of a plane triangle, when a sufficient number of the six parts is given. 3. For the purpose of trigonometrical calculation, the circumference of the circle is supposed to be divided into 360 equal parts, called degrees; each degree is supposed to be divided into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are designated respectively, by the characters". For example, ten degrees, eighteen minutes, and fourteen seconds, would be written. 10° 18' 14". 4. If two lines be drawn through the centre of the circle, at right angles to each other, they will divide the circumference into four equal parts, of 90° each. Every right angle then, as EOA, is measured by an arc of 90°; every acute angle, as BOA, by an arc less than 90°; and every obtuse angle, as FOA, by an arc greater than 90°. 5. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc EB is the complement of AB. The sum of an arc and its complement is equal to 90°. 6. The supplement of an arc is what remains after subtracting the L G E T B H D C supplement of the arc AEF. supplement is equal to 180°. The sum of an arc and its 7. The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, BD is the sine of the arc AB. 8. The cosine of an arc is the part of the diameter intercepted between the foot of the sine and centre. Thus, OD is the cosine of the arc AB. 9. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. Thus, AC is the tangent of the arc AB. 10. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. Thus, OC is the secant of the arc AB. 11. The four lines, BD, OD, AC, OC, depend for their values on the arc AB and the radius OA; they are thus designated : Let the lines ET and IB 12. If ABE be equal to a quadrant, or 90°, then EB will be the complement of AB. be drawn perpendicular to OE. Then, ET, the tangent of EB, is called the cotangent of AB; In general, if A is any arc or angle, we have, L G H 13. If we take an arc, ABEF, greater than 90°, its sine will be FH; OH will be its cosine; AQ its tangent, and OQ its secant. But FH is the sine of the arc GF, which is the supplement of AF, and OH is its cosine; hence, the sine of an arc is equal to the sine of its supplement; and the cosine of an arc is equal to the cosine of its supplement.* E Furthermore, AQ is the tangent of the arc AF, and OQ is its secant: GL is the tangent, and OL the secant of the supplemental arc GF. But since AQ is equal to GL, and OQ to OL, it follows that, the tangent of an arc is equal to the tangent of its supplement; and the secant of an arc is equal to the secant of its supplement.* TABLE OF NATURAL SINES. 14. Let us suppose, that in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent, have been calculated for every minute or second of the quadrant, and arranged in a table; such a table is called a table of sines and tangents. If the radius of the circle is 1, the table is called a table of natural sines. A table of natural sines, therefore, shows the values of the sines, cosines, tangents, and cotangents of all the arcs of a quadrant, which is divided to minutes or seconds. If the sines, cosines, tangents, and secants are known for arcs less than 90°, those for arcs which are greater can be found from them. For if an arc is less than 90°, its supplement will be greater than 90°, and the numerical values of these lines are the same for an arc and its supplement. Thus, if we know the sine of 20°, we also know the sine of its supplement 160°; for the two are equal to each other. The Table of Natural Sines, beginning at page 63, of the tables shows the values of the sines and cosines only. *These relations are between the numerical values of the trigonometrical lines; |