Imágenes de páginas
PDF
EPUB

We may now check the results by computing the values of

[blocks in formation]

Although the agreement is not perfect, the deviations are no greater than those due to the omission of decimals in the logarithms and angles. The angle B is that for which the check is most doubtful, because it is so near 90° that it may be changed 1' or 2' without changing the last figures in log. sin. by more than one or two units in the fifth place. In such a case certainty can be reached only by a duplicate computation.

EXERCISES.

1. Given a 105° 6′.8, b = 93° 39′.9, c = 50° 20′.3; find the angles.

2. Given A = 46° 59′.3, B = 122° 32′.6, C = 139° 0'.3; find the sides.

3. Given A 78° 40'.7, B = 102° 29'.5, C 86° 49'.4; find = = the sides.

4. If the sides of a spherical triangle are each 120°, find the following numerical expressions for the sine, cosine, and tangent of each of the angles, and thence, by the aid of the tables, find the angles themselves.

[blocks in formation]

121. Cagnoli's equation. Cagnoli's equation is

sin a sin b + cos a cos b cos C = sin A sin B-cos A cos B cos c. (15) Proof. Multiplying the third equation (§ 97, 1) by cos C, we

have

cos c cos C = cos a cos b cos C+ sin a sin b cos' C

= cos a cos b cos C+ sin a sin b - sin a sin b sin2

• C. } (a)

The equation of sines (§ 99, 2) gives

[blocks in formation]

whence, by multiplying,

sin a sin b sin' C = sin A sin B sin' c

sin A sin B-sin A sin B cos' c.

Substituting this value in (a), and interchanging the terms of the equation,

sin a sin b+cos a cos b cos C

= cos c cos C+ sin A sin B – sin A sin B cos' c = sin A sin B + cos c (cos C- sin A sin B cos c). From the last equation (11), § 104, we have

cos Csin A sin B cos c =

cos A cos B.

Making this substitution, we have the equation (15) as enunciated. Of course two other similar equations may be obtained by permuting the symbols.

122. Gauss's equations. We write the four equations,
(a) Cagnoli's equation,

(b) and (c) the fundamental equations (1), and (3),, and
(d) the identical equation 1 = 1, as follows:

(a) sin a sin b + cos a cos b cos C = sin A sin B — cos A cos B cos c; (b) cos a cos b+ sin a sin b cos C = cos c;

(c)

(d)

cos C =
1 = 1.

cos A cos B+ sin A sin B cos c;

Taking the sum of the four equations, and substituting cos (a - b) = cos a cos b + sin a sin b,

we have

cos (A+B) = cos A cos B sin A sin B,

cos (ab)+cos C cos (a - b) + cos C+1=

or

---

cos (A+B)

cos c cos (A+B) + cos c + 1;

(1+cos C) [1 + cos (a - b)] = (1 + cos c) [1 - cos (A+B)] (e)

-

If we form the sum — (a) — (b) + (c)+(d), and reduce in the same way, we find

(1+cos C) [1-cos (a - b)] = (1 — cos c) [1 - cos (A - B)] (ƒ) The sum (a)+ (b) − (c) + (d) gives

[ocr errors]

(1

-cos C) [1+ cos (a+b)] = (1 + cos c) [1 + cos (A+B)] (g) The sum (a) (b) — (c) + (d) gives

[ocr errors]

(1 - cos C') [1 - cos (a+b)] = (1 cos c) [1 + cos (A — B)] (h) In the equations (e), (f), (g), (h) we substitute the values of 1±cos, namely,

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

In strictness, the second members of this equation may have the negative as well as the positive sign. But it is easy to show that when the sides and angles are all less than 180°, all the members of the equations are positive. Hence the positive sign is the only one necessary to be taken into account.

These equations are applicable when any three consecutive parts of the triangle-two angles and the included side, or two sides and the included angle—are given.

In the first case the three given parts are all in the right-hand members of the equations; in the second case they are all on the left.

These equations are written in the most convenient order for use in the first case; in the second, the following is the order and arrangement:

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

These are commonly known as Gauss's equations, after Gauss, who first introduced them into astronomical computations. They had, however, been previously published anonymously by Delambre. Example. Given a 132° 46'.7, b = 59° 50′.1, C = 56° 28'.4, to find the remaining parts.

[blocks in formation]

Compute the remaining three parts of each of the following triangles by Gauss's formula:

[blocks in formation]

123. Napier's analogies. If, in the preceding problem, only the two remaining sides in the one case, or the two remaining angles in the other, are wanted, the process may be shortened. Dividing the first of (16) by the second, and the third by the fourth, we have

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

from which may be found a and b when A, B, and C are given. In the same way we find, from (17),

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

from which A and B may be found when a, b, and Care given.

The equations (18) and (19) are known as Napier's analogies.

« AnteriorContinuar »