72. Derivation of a formula for the sine of half an arc in terms of the 73. Derivation of a formula for the sine of half an angle in terms of the three sides of a spherical triangle, and example of its application, 73 74. Derivation of formulæ for the sum and difference of the sires of two 75. Two sides and the included angle being given, the method of solu- 77. Definitions and general principles, 78. The hypothenuse of a right-angled triangle, being either given or 81. Two of the three given parts being a side and its opposite angle, 82. Formula for the cosine of an angle in terms of the three sides, 83. Formula for the difference of the cosines of two arcs, 84. Three sides of a triangle being given to find the angles, with an 85. Three angles being given to find the sides, 86. Formulæ for the sum of the cosines of two arcs, their ratio to the sum of the sines, to the difference of the sines, that of the sine of the sum to the sum of the sines and that of the sine of an arc to radius 88. Napier's rules for the circular parts, 89, 90 and 91. Astronomical examples, 92. Example, given the sun's declination to find the time of his rising or setting, with explanation of the difference between mean and 105. Corrections to be applied to the observed altitudes of celestial objects, 152 106. Dip or depression of the horizon, 111. Method of determining the latitude at sea by the meridian altitude, 161 113. On finding the longitude by the lunar observations, 115. Formulæ for the tangent of the sum and difference of two arcs, 116. Value of the sine and cosine of 45°, 117. Sin an arc = chord of the arc, 118. Sine of 30° and secant of 60°, 120. Generallization of formulæ for the sine of the sum of two arcs, 123. Solution of certain cases of plane triangles and the trigonometrical 125. Formulæ to be employed instead of Napier's rules in certain cases where great accuracy is required, 126. Two sides and the included angle of a spherical triangle being given to find the third side directly, 127. Two angles and the interjacent side being given to find the third 128. Rules relative to ambiguous cases, 129. Additional formulæ where three sides or three angles of a spherical 204 205 PART I. PLANE TRIGONOMETRY. Ꭲ 1. The term TRIGONOMETRY is compounded of two Greek words toyovos a triangle, and uerpov measure, signifying literally the measurement of triangles. It has for its object to determine the unknown parts of a triangle when a sufficient number of parts are known. By parts of a triangle are understood commonly the sides a and angles, though trigonometry properly includes the measurement of the surface also. There will accordingly be six parts of every triangle, namely the three sides and the three angles. 2. It has been proved, (Geom. *B. 1, Props. 5, 6, and 10,) that when two triangles have three parts, one of which is a side, in the one equal respectively to the corresponding parts in the other, the triangles are equal. One part must be a side, because if the three angles only were equal respectively in the two triangles they would be but similar, (Geom. B. 4, Prop. 18,) that is alike in shape but not necessarily in size. Since all triangles which have three parts equal, are by consequence equal, it is said that three given parts determine a triangle, that is with these three given parts but one triangle can be formed. If any number of attempts be made to form a new triangle with the same three given parts, the result will be always a repetition of the same triangle. The corresponding sides of the successive ones will not differ in length, and the angles will not differ in magnitude. There is one exception to this principle, pointed out in B. 3, Prob. 11, Geom., where two sides and the angle opposite one of them are given, in which case two triangles can be constructed with the given parts. 3. Three parts of a plane triangle being given then, (except they be the three angles,) it ought to be possible to find the other three, since these are fixed by their dependance upon the three given. This may be accomplished with sufficient accuracy for many purposes, by means of constructions, such as are exhibited at problems 8, 9 and 10 of the 3d book of Geometry. We shall repeat one of these constructions, enunciating the problem somewhat differently. The two sides and included angle of a triangle being given, let it be required to find the remaining side and the other two angles. A B Let A and B be the two given sides, and c the given included angle. Draw two lines DH and G of indefinite length, making with each other an angle equal to the given angle c. Lay off on the first of these the given linea D from D to E, and on the second the given line B from p to F. Join EF. The only possible triangle DEF will thus be formed with the three given parts, in which EF will be the required side, and E and F the required angles. The finding the unknown parts of a triangle by means of those which are given, is called its solution. 4. The method of solution just exhibited is rendered more practically useful by the employment of scales of equal parts and protractors. В H E В a The most simple form of a scale of equal parts, is shown in the annexed figure. 9 8 7 6 5 4 3 2 1 It is a straight rule divided into any number of equal parts: in this example ten and one of these again into ten, so that the smallest division is one hundredth of the whole length of the rule. The following is the manner of using it. Suppose that it is required to draw upon paper a line equal in length to 56. Place one foot of a pair of dividers at the line of division marked 5, and extend them till the other foot reaches exactly to the sixth smaller division mark on the right of 0; the feet of the dividers will then be at a distance of 56 apart. To draw now the required line upon paper, let a be the point from which it is to be drawn. Placing one foot of the dividers at A, extended the distance 56 obtained from the scale, describe with the other an arc of a circle on the side towards which the line is to be drawn; then from a draw the line in the proper direction, terminating it at the arc before described, and it will be the line required. Another line of 42 being measured from the scale and laid down upon the paper, the two lines will be in the ratio of 56 to 42. If they are lines upon a map, and the first corresponds to a line of 56 feet upon the ground, the second will correspond to a line of 42 feet. If the first represent 56 yards, or chains, or miles, the second will represent 42 yards, or chains, or miles. And in general lines upon the same drawing which are measured in parts of the same scale must be understood to be expressed in units of the same kind. 5. Before describing the protractor which is an instrument for laying off angles, it will be necessary to explain the method of estimating the magnitude of angles. |