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sin. A sin. B:: a:b

.*. sin. A + sin. B: sin. Asin. B:: a+ba-b;

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that is to say, in any plane triangle the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference.

By help of this rule we may determine the remaining parts of the triangle, when we know two sides a, b, and the included angle C; for knowing C we know also † (A + B) = (180° — C); and † (A — B) is determined by this rule; therefore, as the half sum added to the half difference of two quantities gives the greater, and subtracted gives the less; we thence readily obtain the angles A and B, and then the third side c, by (16).

We have thus deduced commodious rules fitted for logarithmic computation, for the solution of the first two cases of plane triangles: it remains to furnish a rule for the third case.

(20.) Referring to the expression for cos. A at (17), it is plain that

since

and, therefore,

b2 + c2= (b+c)2 - 2 bc,

b2 + c2 — a2 = (b+c+ a) (b + c − a) →2bc;

that expression may be put under the form

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Now supposing the arcs a, a', in equation (A), to be equal to each other, and to A, we have from the second of them

cos. A cos.2 A-sin.2 A

1= cos.2 A+ sin.2 ¦ A

by addition, cos. A=2 cos.2 A ·

by subtraction, cos. A=1-2 sin.2 A;

by substituting the first of these values in the foregoing equation, and putting for brevity S for the sum of the three sides of the triangle, we

have

cos. A

S ( S − a)
bc

(1).

We can just as readily obtain a second formula by means of the other expression for cos. A; for substituting it in equation (2), art. (17), we

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and by dividing this expression by the former we get a third formula,

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(21.) We thus have three distinct formulas for the determination of the angles of a triangle when the three sides are given, and all of them are adapted to logarithmic computation. It is not, however, always a matter of indifference which of these formulas we employ, as in certain cases one may be preferable to another. Thus, if we knew beforehand, or could foresee that the sought angle A would be very nearly equal to 90°, then it would be improper to employ the formula (2), because we should be very likely to commit error in taking out the angle, seeing that for an angle very near 90° the seven first decimals in the sine coincide with those in the sines of several other angles in its vicinity, or which differ each from the proposed angle by only a few seconds.

If the logarithmic tables, which we employ, are calculated to seconds, as the large tables, of Taylor or of Bagay, then the sought angle when near 90°, may be accurately determined to the nearest second, either from its cosine or from its tangent, as the values of these trigonometrical lines, at this part of the table differ considerably from each other, even when the

arcs are nearly equal. But if the table employed is not calculated to seconds, then the sought angle, when near 90°, should be determined from its cosine, and not from its tangent; because in approaching to 90° the tangents increase by very unequal differences, and, as in proportioning for the seconds, we proceed on the supposition that the tangents increase equally through 60′′, we shall be in danger of committing error in thus determining the seconds. As the cosines decrease more regularly towards the extremity of the quadrant than the tangents increase, it will, therefore, be safest to determine such arcs from their cosines.

When the sought angle is very small it will be best to determine it from its sine; although the tangent may be used with safety.

Solution of Plane Triangles in general.

(22.) We shall now proceed to apply the rules and formulas which we have just investigated to the several cases of plane triangles, repeating the rule at the head of each case.

CASE I.

When a side and its opposite angle are among the given parts.

RULE.

Sine of given angle,

: its opposite side

:: sine of any other angle

: its opposite side.

Also, any given side,

: sine of its opposite angle

:: any other side

: sine of its opposite angle.

As the same sine belongs both to an angle and to its supplement, it may

seem doubtful in determining an angle of a triangle from its sine, whether to take the acute angle given by the tables or the obtuse angle which is its supplement.

The following precepts will remove all doubt on this point.

1. If the given angle is obtuse the sought angle must be acute. This is obvious, because a triangle cannot have two obtuse angles.

2. If the given angle be acute, and the side opposite to it greater than the side opposite to the sought angle, this must be acute; for the greater angle must be opposite to the greater side.

3. But when the side opposite to the given angle is less than that opposite to the sought angle, this may be either acute or obtuse, so that two triangles exist under the proposed conditions, and the problem in question admits, therefore, of two solutions. The annexed diagram shows that with two given sides AC, CB, and the acute angle A, opposite to one of them, we may always construct two triangles, ABC, AB'C; where the angle B, opposite to the other given side in the one triangle, will be the supplement of the corresponding angle B' in the other, provided CB is less than CA.

B

EXAMPLES.

(23.) 1. In the triangle ABC are given

AB137, AC=153, B=78° 13',

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The obtuse angle, which is the supplement of this, is not admissible, because the side opposite to the given angle is greater than the side opposite the required one.

D

II. To find the side CB.

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The angle A is equal to 180° (B+ C)=180°-139°26′47′′ = 40°33′13′′, therefore,

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2. In the plane triangle ABC are given

AC 216, CB=117, A=22° 37',

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The angle B is, in this example, ambiguous, because the side opposite the given angle is less than that opposite the required one.

II. To find the third side AB.

The angle C is equal to 180° — (A + B) = 112° 9′ 5′′, provided we take B acute; therefore,

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