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complement of the sine and cosine, in the following simple manner :
113. For the arithmetical complement of the sine, subtract 10 from the index of the cosecant; and for the arithmetical complement of the cosine, subtract 10 from the index of the secant.
By this, we may save the trouble of taking each of the figures from 9.
SOLUTIONS OF RIGHT ANGLED TRIANGLES,
Art. 114. In a triangle, there are six parts, three sides, and three angles. In every trigonometrical calculation, it is necessary that some of these should be known, to enable us to find the others. The number of parts which must be given, is THREE, one of which must be a side.
If only two parts be given, they will be either two sides, a side and an angle, or two angles; neither of which will limit the triangle to a particular form and size.
If two sides only be given, they may make any angle with each other; and may, therefore, be the sides of a thousand different triangles. Thus the two lines a and b (Fig. 7.) may belong either to the triangle ABC, or ABC', or ABC". So that it will be impossible, from knowing two of the sides of a triangle, to determine the other parts.
Or, if a side and an angle only be given, the triangle will be indeterminate. Thus, if the side AB (Fig. 8.) and the angle at A be given ; they may be parts either of the triangle ABC, or ABC', or ABC".
Lastly, if two angles, or even if all the angles be given, they will not determine the length of the sides. For the triangles ABC, A'B'C', A"B"C", (Fig. 9.) and a hundred others which might be drawn, with sides parallel to these, will all have the same angles. So that one of the parts given must always be a side. If this and any other two parts, either sides or angles, be known, the other three may be found, as will be shown, in this and the following section.
115. Triangles are either right angled or oblique angled. The calculations of the former are the most simple, and those which we have the most frequent occasion to make. A great portion of the problems in the mensuration of heights and disiances, in surveying, navigation and astronomy, are solved by rectangular trigonometry. Any triangle whatever may be divided into two right angled triangles, by drawing a perpendicular from one of the angles to the opposite side.
116. One of the six parts in a right angled triangle, is always given, viz. the right angle. This is a constant quantity; while the other angles and the sides are variable. It is also to be observed, that, if one of the acute angles is given, the other is known of course. For one is the complement of the other. (Art. 76, 77.) So that, in a right angled triangle, subtracting one of the acute angles from 90° gives the other. There remain, then, only four parts, one of the acute angles, and the three sides to be sought by calculation. If any two of these be given, with the right angle, the others may be found.
117. To illustrate the method of calculation, let a case be supposed in which a right angled triangle CAD (Fig. 10.) has one of its sides equal to the radius to which the trigonometrical tables are adapted.
In the first place, let the base of the triangle be equal to the tabular radius. Then, if a circle be described, with this ra: dius, about the angle C as a center, DA will be the tangent, and DC the secant of that angle. (Art. 84, 85.) So that the radius, the tangent, and the secant of the angle at C, constitute the three sides of the triangle. The tangent, taken from the tables of natural sines, tangents, &c. will be the length of the perpendicular; and the secant will be the length of the hypothenuse. If the tables used be logarithmic, they will give the logarithms of the lengths of the two sides.
In the same manner, any right angled triangle whatever, whose base is equal to the radius of the tables, will have its other two sides found among the tangents and secants. Thus, if the quadrant AH (Fig. 11.) be divided into portions of 15° each ; then, in the triangle
CAD, AD will be the tan, and CD the sec of 15°, In-CAD', AD' will be the tan, and CD' the sec of 30°, In CAD", AD" will be the tan, and CD" the sec of 45°, &c.
118. In the next place, let the hypothenuse of a right angled triangle CBF (Fig. 12.) be equal to the radius of the ta. bles. Then, if a circle be described, with the given radius, and about the angle C as a center; BF will be the sine, and BC the cosine of that angle. (Art. 82. 89.) Therefore the sine of the angle at C, taken from the tables, will be the length
of the perpendicular, and the cosine will be the length of the base.
And any right angled triangle whatever, whose hypothenuse is equal to the tabular radius, will have its other two sides found among the sines and cosines. Thus if the quadrant AH (Fig. 13.) be divided into portions of 15° each, in the points F, F, F", &c. ; then, in the triangle,
CBF, FB will be the sin, and CB the cos, of 15°, In CB'F', F'B' will be the sin, and CB' the cos, of 30°, In CB"F", F"B" will be the sin, and CB" the cos, of 45°, &c.
119. By merely turning to the tables, then, we may find the parts of any right angled triangle which has one of its sides equal to the radius of the tables. But for determining the parts of triangles which have not any of their sides equal to the tabular radius, the following proportion is used:
As the radius of one circle,
degrees, in the other. In the two concentric circles AHM, ahm, (Fig. 4.) the arcs AG and ag contain the same number of degrees. (Art. 74.) The sines of these arcs are BG and bg, the tangents AD and ad, and the secants CD and Cd. The four triangles, CAD, CBG, Cad, and Cbg, are similar. For each of them, from the nature of sines and tangents, contains one right angle ; the angle at C is common to thein all; and the other acute angle in each is the complement of that at C. (Art. 77.) We have, then, the following proportions. (Euc. 4. 6.)
1. CG : Cg::BG : bg. That is, one radius is to the other, as one sine to the other.
2. CA : Cu::DA: da.
That is, one radius is to the other, as one tangent to the other.
3. CA : Ca::CD : Cd. That is, one radius is to the other, as one secant to the other.
Cor. BG : bg::DA : da::CD : Cd.
That is, as the sine in one circle, to the sine in the other ; so is the tangent in one, to the tangent in the other; and so is the secant in one, to the secant in the other.
This is a general principle, which may be applied to most trigonometrical calculations. If one of the sides of the proposed triangle be made radius, each of the other sides will be the sine, tangent, or secant, of an arc described by this radius. Proportions are then stated, between these lines, and the tabular radius, sine, tangent, &c.
120. A line is said to be made radius, when a circle is described, or supposed to be described, whose semi-diameter is equal to the line, and whose center is at one end of it.
121. In any right angled triangle, if the HYPOTHEnuse be made radius, one of the legs will be a sine of its opposite angle, and the other leg a cosine of the same angle.
Thus, if to the triangle ABC (Fig. 14.) a circle be applied, whose radius is AC, and whose center is A, then BC will be the sine, and BA the cosine, of the angle at A. (Art. 82, 89.)
If, while the same line is radius, the other end C be made the center, then BA will be the sine, and BC the cosine, of the angle at C.
122. If either of the legs be made radius, the other leg will be a tangent of its opposite angle, and the hypothenuse will be a secant of the same angle; that is, of the angle between the secant and the radius.
Thus, if the base AB (Fig. 15.) be made radius, the center being at A, BC will be the tangent, and AC the secant, of the angle at A. (Art. 84, 85.)
But, if the perpendicular BC (Fig. 16.) be made radius, with the center at C, then AB will be the tangent, and AC the secant, of the angle at C.
123. As the side which is the sine, tangent, or secant of one of the acute angles, is the cosine, cotangent, or cosecant of the other ; (Art. 89.) the perpendicular BC (Fig. 14.) is the sine of the angle A, and the cosine of the angle C ; while the base AB is the sine of the angle C, and the cosine of the angle A.
If the base is made radius, as in Fig. 15, the perpendicular BC is the tangent of the angle A, and the cotangent of the angle C; while the hypothenuse is the secant of the angle A, and the cosecant of the angle C.
If the perpendicular is made radius, as in Fig. 16, the base AB is the tangent of the angle C, and the cotangent of the