We may

also find c directly by geometry, from

c2 = a2 + b2 whence c = √(a2 + b2)

but this is not readily computed by logarithms.

EXAMPLES.

1. Given a = 30, b = 40; find c.

2. Given a = 8.678912, b = 2.463878; find A and c.

Ans. c = 50

ADDITIONAL FORMULE FOR RIGHT TRIANGLES.

112. By inspecting the tables it will be seen that when the angles are very small, the cosines differ very little from each other; consequently a small angle cannot be found with very great accuracy from its cosine. For a similar reason an angle that is nearly 90° cannot be accurately computed from its sine. It is therefore desirable, when a required angle is small, to find it by its sine, and when near 90° by its cosine, or in either case by its tangent or cotangent; and for this purpose special formulæ are sometimes necessary. We shall deduce several such formulæ, from which one adapted to a particular case may be selected.

which may be used instead of (197), when A is small, that is when b is nearly equal to c. It gives also

which, since 2 A = A + 90° — B = 90° — (B — A), gives

by which B-A is found with great accuracy when b and a are nearly equal. EXAMPLE. Given c 4602.836, b = 4602-21059 to find A.

The ordinary process gives log cos A = 9.9999410, whence A 56' 40". These results are obtained by Stanley's Tables, in which the log. sines, &c., are given for every 10" for the first 15°. A greater discrepancy between the two results would be found by tables in which the functions were given only for each minute. A slight error remains in the value of ≥ A : differences of the log. sines in this part of the

= 28′ 20′′-18, on account of the large

table, or rather on account of the

rapid change of these differences. We avoid the use of these large differences, and gain somewhat in accuracy, by employing the approximate value of sin. A given by (98), whence

But to obtain A with the utmost precision, recourse must be had to the following process, which is constantly employed in observatories, and wherever small angles are to be computed with extreme accuracy. Special tables are prepared containing for every minute from 0° to 2° the logarithms of

which do not vary rapidly, and may therefore be taken with accuracy from the tables. Then we have

A table of this kind will be found on page 156 of Stanley's Tables, where the notation used is

sin x

and therefore in the column marked q―n we find the log

Thus in the above

the use of which may easily be inferred from the example just given.

CHAPTER VII.

FORMULÆ FOR THE SOLUTION OF PLANE OBLIQUE TRIANGLES.

116. As every oblique triangle may be resolved into two right triangles by a perpendicular from one of the angles upon the opposite side, we are enabled to deduce all the formula for their solution from those of the preceding chapter.

117. The sides of a plane triangle are proportional to the sines of their opposite angles.

Denote the angles of the triangle ABC, Fig. 16, by A, B and C, and the sides opposite these angles respectively by a, b and c. From C draw CP perp. to AB and put

b

Fig. 16.

C

B

P

CP p. Then in the right triangles ACP, BCP, we have,

=

=

and these three proportions may be written as one, thus:

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

:

When the perpendicular falls without the triangle, Fig. 17, the angle CBP is the supplement of B, but by Art. 39, it has the same sine, so that the triangle CBP gives

(217)

(218)

Fig. 17.

P

B

the same as was found from Fig. 16. The proposition is therefore general in its application.*

118. The sum of any two sides of a plane triangle is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference.

But from (109) if x = A, y = B we obtain the proportion sin A+ sin B: sin Asin B = tan (A + B): tan (A — B) which, compared with the above, gives

and we may infer the same relation between b, c, B, C and a, c, A, C.

119. The square of any side of a triangle is equal to the sum of the squares of the other two sides diminished by twice the rectangle of these sides multiplied by the cosine of their included angle.

In the triangle ABC, Figs. 16 and 17,* we have either

The consideration of Fig. 17 was not strictly necessary according to the principle stated in Art. 49. It may, however, be useful for the student to verify that principle when convenient.