ADC, AC 288, ACD 30° 6′ ... A=90° — 39° 6′ = 50° 54′;

hence, to find AD, we have by making AC radius,

A = 54° 36′ 14′′, C=35° 23′ 46′′, AC=72.23.

5. Given the hypotenuse AC=643 · 7, and the base AB=473·8, to find the other parts?

A = 42° 36′ 12′′, C=47° 23′ 48′′, BC=35·87. 6. Given the angle A= 37° 2′ 43′′, and the hypotenuse AC=173·2 to find the other parts?

C=52° 57′ 17′′, AB= 138.24, BC=104·34.

(16.) We shall now proceed to investigate rules and formulus for the solution of triangles in general.

Oblique-Angled Triangles.

Let ABC be any plane triangle, and let us denote the angles by the capital letters, A, B, C, at their vertices, and the sides opposite to them by the small letters a, b, c.

e

RA

'D

From either vertex, as C, draw the perpendicular CD to the opposite side. Then the sine of A to the radius b will, obviously, be the line CD, and the A value of this sine in terms of the trigonometrical sine of the same angle to radius 1 is (art. 5,) CD = b sin. A. In like manner the sine of B to the radius a, is the same line CD, whose value, therefore, in term of the trigonometrical sine, is CD = a sin. B; consequently, by

This equation immediately furnishes us with an important rule, which may be expressed as follows.

Any side of a triangle is to any other side as the sine of the angle, opposite to the former, is to the sine of the angle opposite to the latter. Whenever, therefore, we know two sides and an angle opposite to one of them, or two angles and a side opposite to one of them, the other three parts of the triangle may always be determined by help of this rule.

The cosine of A, to the radius b, is the line AD; and, therefore, AD, in terms of the trigonometrical cosine of A, is AD=b cos. A. In like manner the cosine of B to the radius BC, is BD, which, in terms of the trigonometrical cosine, is BD = a cos. B; if the angle B is obtuse, as in the second of the above diagrams, cos. B will be negative; hence whether it be acute or obtuse we shall have for the side AB the expres

sion

ca cos. Bb cos. A;

in which the proper signs of the cosines are supposed to be involved in their expressions.

If instead of drawing the perpendicular from C we had drawn it from B, it is easy to see the result we should have obtained; for then considering B the vertical angle instead of C, or supposing the triangle to be turned about till B actually becomes the vertical angle, then commencing at the vertex, the arrangement of the angles will now be B, C, A ; these, therefore, should respectively be substituted for C, A, B, in the above formula; also the arrangement of the sides will be a, b, c, instead of b, c, a, as at first, so that these letters must be replaced by the former: consequently, our equation will become

b=c cos. A+ a cos. C.

If, on the contrary, A be made the vertical angle, then the order of the angles will be A, B, C, and of the sides c, a, b, and these must supply the places, of C, A, B, and b, c, a, in the first formula, so that we shall

and these equations contain the whole theory of plane trigonometry. They involve all the six parts of a triangle, the three angles, and the three sides; and, as the equations are three in number, any three of the parts, considered as unknown quantities, may be determined, provided only the other three are known; but fewer than three being given will not be sufficient to determine the others, as then there would be a greater number of unknowns than of equations.

We must remark too that the three given quantities must not be the three angles simply, because the three other quantities a, b, c, severally enter the three terms of each equation, so that if we were to multiply each equation, by any assumed factor whatever, m, the values resulting from the elimination of A, B, C, would, obviously, be the same for ma, mb, mc, as for a, b, c; thus, showing that the data are not sufficient to determine any triangle, but belong equally to innumerable triangles, all, however, similar to each other.

(17.) It appears then that the solutions to all the cases of plane triangles are derivable from the equations (1), under different hypotheses, as to the three unknown quantities, and we might now with but little trouble proceed to deduce these solutions, one after another, from these equations: thus suppose the three sides a, b, c, were given, then multiplying the first equation by a, the second by b, and the third by c, we have

a2=ab cos. C + ac cos. B

b2bc cos. A + ba cos. C

c2 ac cos. B + bc cos. A;

and subtracting each of these from the sum of the other two, we get

and thus the values of the cosines of the required angles become known, and by searching in the table of natural sines and cosines we shall find the angles to which they belong.

It is necessary to remark here that in almost every trigonometrical calculation it is advisable to conduct the operation by means of logarithms, in order to avoid lengthy and tiresome multiplications, divisions, and extractions; so that it becomes a matter of consequence to express all our general rules and formulas in a form, adapted as much as possible to logarithmic calculation, that is, the operations indicated by the formulas should be those of multiplication, division, involution, and evolution, and not those of addition and subtraction.

The formulas just deduced for the angles of a triangle, when the sides are given, do not appear in a form adapted to logarithmic computation; and the same would be found to be the case with the various other formulas directly deducible from the general equations (1); nor would it be easy, without the aid of other and independent properties, to convert these expressions into the desired form. Although, therefore, it is true, as we have stated above, that formulas for all the cases of plane trigonometry may be deduced from the equations (1), yet, on account of the inconvenient form these formulas assume, it becomes necessary for us to seek assistance from other sources. Now there exists two general trigonometrical formulas, which may be considered as forming the foundation of the whole theory of angular magnitude, and which, in conjunction with what is laid down above, will enable us to deduce formulas suited to logarithmic calculation for all the cases of plane triangles.

(18.) There are various ways of investigating these formulas; we shall adopt that which appears to us the most simple and general.

It was given by M. Sarrus in the Annales des Mathématiques,

tom. xi.

Given the sines and cosines of two arcs or angles, to find the sine and cosine of their sum and difference.

Let AM=a, and AN=a', be any two arcs of the circle, the radius being unity, then drawing the chord of the arc NMaa', we shall have from the triangle NMG right angled at G,

MN2 = NG2 + MG2 = (CQ — CP)2 + (PM — NQ)2;

which may be written thus,

chd.2 (a — a') = (cos. a′ — cos. a)2 + (sin. a By actually squaring the expressions in the right-hand member of this equation, and recollecting that

sin.2 a + cos.2 a =1, sin.2 a' + cos.2 a′ =1,

we have

– sin. a′)2.

chd.2 (aa)=2—2 cos. a cos. a -2 sin. a sin. a'

Suppose now that a=0, then we have

chd.2 a 2-2 cos. a." *

N

· (1).

As this expression is true for any arc whatever, it is true for the arc - a', so that

Comparing together the second members of (1) and (2) we obtain

cos. (aa)=cos. a cos. a' + sin. a sin. a' . . . . (1). As this is true for all values of a, a', it is true when a — a', so that

Cos. a' Cos. a cos. (a — a') + sin. a sin. (a — a');

a' is put for

in which equation, if we substitute the value of cos. (a — a′) given by (1), we have

cos. a'cos. a2 cos. a' + cos. a sin. a sin. a' + sin. a sin. (a — a');

* This property is also proved in the Geometry, p. 92, Scholium.