is the cosine, CI the cotangent, and II the cosecant of AB. From the definitions it is evident, that the sine, tangent, and secant, are common to two arcs, which are the supplements of each other. So the sine, tangent, or secant, of 50° is also the sine, tangent, or secant of 130°. The sine, tangent, or secant, of an angle is the sine, tangent, or secant of the arc, or the degrees, by which the angle is measured. The sine, tangent, and sccant of every degree and minute in a quadrant are calculated to the radius 1, and ranged in tables for use. But because trigonometrical operations with these natural sines, tangents, and secants require tedious multiplications and divisions, the logarithms of them are taken, and ranged in tables also ; and the logarithmic sines, tangents, and secants are commonly used, as they require only additions and subtractions, instead of the multiplications and divisions. There are usually three methods of resolving triangles, or the cases of trigonometry ; namely, Geometrical Construction, Arithmetical Computation, and Instrumental Op , eration, In the first method ; let the triangle be constructed by making the parts of the given magnitudes, namely, the sides from a scale of equal parts, and the angles from a scale of chords, or other instrument. Then measure the required parts by the same scale. In the second method ; having stated the terms of the proportion according to the rule, resolve it like all other proportions, in which a fourth term is to be found from three given terms, by multiplying the second and third together, and dividing the product by the first, in working with the natural natural numbers, whether they be sides, or sincs, tangents, or secants, of angles. Or, in working with logarithms, add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first term ; then the number answering to the remainder will be the fourth term required. To work a stating instrumentally ; as, for example, by the logarithmic lines on one side of the two-foot scales.--Extend the compasses from the first term to the second, or third, which happens to be of the same kind with it ; then that extent will reach from the other term to the fourth, taking both extents toward the same side. Note. For the sides of triangles the line of numbers, marked Num. is used ; and for the angles, the line of sines, or of tangents, marked Sin. or Tan. according as the proportion respects sines or tangents. If the extent upon the tangents reach beyond the line, set it so far back as it reaches over. In a triangle there must be given three parts, one of which, at least, must be a side ; because the same angles are common to an infinite number of triangles. In plane trigonometry, there are only three cases, or varieties, viz. 1 1. When two of the three given parts are a side and its opposite angle. 2. When two sides and their included angle are given. 3. When the three sides are given. PROBLEM Given three such parts, that an angle and its opposite side shall be two of them; to find the rest. In any plane triangle, the sides are proportional to the sines of their opposite angles.* That is, As one side NOTE 1. To find an angle, begin the proportion with a side opposite to a given angle ; and to find a side, begin with an angle opposite to a given side. NOTE 2. An angle, found by this rule, is always ambiguous, except it be a right angle, or except, that the magnitude of the given angle prevent the ambiguity ; because the sine answers to two angles, which are the sup plements * DEMONSTRATION. and upon Let ABC be B AC demit the perpendiculars DF, EG, BH; then will DF and EG be the sines of the A. aogles A, C, to the general radius AD or CE. Now from similar triangles we shall BH AD DF BH AD (CE) :: and hence, of equality, AB : BC EG ; DF. have {CE EG} : plements of each other : and accordingly the construction gives itwo triangles with the same given parts; and when there is no restriction or limitation, included in the propo. sition, either of them may be taken. The degrees, in the table, answering to the sine, is the acute angle ; and if the angle be obtuse, take those degrees from 180°, and the remainder will be the obtuse angle. When the given angle is obtuse, or right, there can be no ambiguity ; for then neither of the other angles can be obtuse, and the construction will produce but one triangle. 1. Draw the line AB=345 from some convenient scale of equal parts: 2. Make the angle A=37° 20'. 3. With the centre B, and radius 232, taken from the sanie scale of equal parts, cross AC in C. 4. Draw BC, and the triangle is constructed. Then Then the angles B and C, measured by the scale of chords, and the side AC, measured by the scale of equal parts, will be found to be as follows : viz. LB 27° LC 1153 1 Or 645 Or 374 AC 174 As side BC 232 To side BA 345 So sine ZA37° 20' 2'3654880 2'5378191 97827958 To sine <C 115o 36' or 64° 24' 9'9551269 ZA 37 20 37 20 In the first proportion, extend from 232 to 345 upon the line of numbers ; that extent will reach, upon the : sines, from 37% to 64 the angle C. |