by which we in general compute the sides and angles of a triangle from certain data. Investigations of the same kind are of no further use to the reader, and we shall therefore give merely the theoretical solutions for the future.

ART. 22. In a right-angled triangle given the side opposite the right angle and one of the remaining sides, the triangle itself may be determined.

Representing the sides and angles as above, then a, b, A, being given, c, B, C, may be determined.

DEM. For c= √a2—b2, by ART. 47, GEOM.; and sin b

B =

b

a

COS C =

*

In logarithms, as

√/a°—b2 = √(a+b). (a−b) = (a+b)3. (a—b)§, .. log. c=log. (a+b)* . (a—b)*=log. (a+b)3+log. (a—b)✯ ; .. by ART. 6, LOG., log. c= log. (a+b) + 1⁄2 log. (a—b). And,

This, &c.

log. b

log. a,

log. sin B =
log, cos c = log. b — log. a.

ART. 23. In a right-angled triangle given the side opposite the right angle, and one acute angle, the triangle itself may be determined.

Representing the sides and angles as above, then a, B, A, C, being given,† b, c, may be determined.

* To shorten computation we will use the ordinary radius 1, instead of 1010, which is only necessary when we would solve the question arithmetically.

+ One acute angle, as B, of a right-angled triangle, being given, both are given, inasmuch as c = 90°-B.

In logarithms,

log. b = log. a + log. sin B, and, log. c = log. a + log. cos B.

This, &c.

ART. 24. In a right-angled triangle given either side about the right angle, and either acute angle, the triangle itself may be determined.

Representing the sides and angles as above, then b, a, B, C, being given, a, c, may be determined.

Triangles which are not right-angled may be solved, that is, determined from certain data, with little less facility than the above.

ART. 25. In any triangle given two sides and the included angle, the triangle itself may be determined.

Let a, b, c be respectively the three sides of the triangle, A, B, C the opposite corresponding angles. Then, any two sides, as a, b, and the included angle c, being known, the third side c, and the remaining angles A, B, may be A determined.

DEM. BY ART. 17,

с

B

a

but as A + B + c = 180o, ... A + B = 180o C,

A+ B 2

C

=900—

2*

Consequently, subtracting half the given angle

c from a right angle, we obtain

A+B
2

and therefore can find

its tangent in the tables. Hence, as a and b are likewise

given, we obtain the whole value of tan

A-B

from whence

> 2

we get the corresponding angle by the tables. But

A+B
?
2

A-B
2

A-B

from half the sum two angles A and B, we can determine both separately; as we have only to add half the difference to half the sum for the greater, and to subtract half the difference from half the

and half the difference, of the

sum for the lesser. Thus, if c = 40°, then = =90o.

A+B
2

C

2

=70°; and if, from the above method of computation, we

Again having thus computed A and B, we may find c. For, by ART. 16,

'A-B'

*It is plain from the formula for tan (4), that, to compute

it, we need not be given the absolute magnitude of the sides a, b, but merely their ratio. For suppose that ratio be m: n,

i. e.

from which formula for c its value may be determined, as a, sin c, and sin A, are known quantities.

ART. 26. Given two angles of a triangle, and the side opposite either of them, the triangle itself may be determined.

Representing the sides and angles as above, then any two angles, as A, B, and a side, as a, opposite either of these angles, A, being given, the third angle, c, and the remaining sides, b, c, may be found.

DEM. The third angle, c, is found by subtracting the sum of A and B from 180°; for A + B + c = 180o, .'. c = 1800 (A + B). Likewise, by ART. 16,

b

sin B

sin B

b=a. 9

a

sin A

sin A

Hence, b and c are found in terms either given as a, or easily computable from trigonometrical tables, as sin a, sin B, sin c.

In logarithms,

log. b = log. a + log. sin B and, log. c = log. alog. sin c This, &c.

log. sin A.

log. sin A,

ART. 27. Given two angles of a triangle, and the side between them, the triangle itself may be determined.

For two angles being given the third is determinable, as in ART. prec.; and the solution of this question will be thus reduced to the last.

ART. 28. Given the three sides of a triangle, the triangle itself may be determined.

Representing the sides and angles as above, then a, b, c being given, A, B, C may be found.

c may be found immediately; because cos c × 2ab = a2 + b2 -c2, by ART. 20. In this formula, a, b, and cos c are known quantities, whence c may be found.

From which value, in known terms, for cos A, we may find A itself from the tables. Hence, the triangle may be determined by ART. 23, as we know one of its angles and the including sides. This, &c.

Obs. We may determine B and C without the help of ART. 23, as we determined A, namely, by the formula (ART. 20).

There are also various other methods of resolving this question, but one is sufficient to our purpose.*

ART. 29. Given two sides of a triangle, and the angle opposite either of them, the triangle itself may be in some cases determined.

Representing the sides and angles as above, then any two sides, as a, b, and the angle, as A, opposite either of them, a, being given, the triangle itself may be determined in two cases, viz. 1o, if the given side not opposite the given angle be less than the other given side; 2o, if the given side not opposite the given angle be less than the third side.

log. sin B = log. sin a log. blog. a. Consequently sin в is determined, the quantities b, a, and sin A, being known. But as, by ART. 6, the sine is the same for two supplementary angles, we cannot determine

* It should be observed, that the method here given is not adapted to logarithms, and therefore others of a more complex nature are used in preference.