Sidebilder
PDF
[subsumed][subsumed][ocr errors]

sin B : cos C =R:cos AB, B and C. AC. sin C: cos B=R:cos AC, 76)

tan B:cot C=R: cos BC, 13) (16

[ocr errors]
[ocr errors]

Table for determining when the things found in the pre

ceding are less than a Quadrant (Sp. Ge. 14 and 15).

The angle or arc found is less than 90°.'
When B is less than 90°.
When BC and B are of the same affection.
When BC and B are of the same affection.
When C is less than 90°.
When AC and C are of the same affection.
When AC is less than 90°.
Ambiguous.
Ambiguous.
Ambiguous.
When AC and BC are of the same affection.
When AC is less than 90°.
When AC and BC are of the same affection.
When AB and AC are of the same affection.
When AC is less than 30°.
When AB is less than 90°.

When C is less than 90°. When B is less than 90°. | When B and Care of the same affection.

16

Schol. The rules for the cases of right-angled spherical trigonometry may be reduced to two, called Napier's Rules of the Circular Parts.

In a right-angled spherical triangle, the right angle is neglected, and the hypotenuse, the two angles, and the complements of the two sides, are called circular parts ; and any of these parts being called the middle part; the two adjacent to it, adjacent parts; and the two remaining parts, opposite parts; then, the rectangle under the radius and the cosine of the middle part, is equal to that under the cotangents of the adjacent parts, or the sines of the opposite parts. Or if the middle part be called M; the two adjacent parts, A and a; and the opposite parts 0 and 0,

Rocos M = cot Ā.cot a,

or Rocos M = sin 0 • sin 0. Either of these rules may be converted into a proportion by Pl. Ge. VI. 16.

The cases marked ambiguous are those in which the thing sought has two values, and may either be equal to a certain angle, or to the supplement of that angle. Of these there are three, in all of which the things given are a side, and the angle opposite to it; and accordingly, it is easy to show that two right-angled spherical triangles may always be found, that have a side and the angle opposite to it the same in both, but of which the remaining sides, and the remaining angle of the one, are the supplements of the remaining sides and the remaining angle of the other, each of each.

Though the affection of the arc or angle found may in all the other cases be determined by the rules in the second of the preceding tables, it may be useful to remark, that all these rules, except two, may be reduced to one, namely, that when the thing found by the rules in the first table is either a tangent or a cosine; and when, of the tangents of cosines employed in the computation of it, one only belongs to an obtuse angle, the angle required is also obtuse.

Thus, in the 15th case, when cos AB is found, if C be an obtuse angle, because of cos C, AB must be obtuse; and in case 16, if either B or C be obtuse, BC is greater than 90°, but if B and C are either both acute, or both obtuse, BC is less than 90°.

It is evident that this rule does not apply when that which is found is the sine of an arc; and this, besides the three ambiguous cases, happens also in other two, namely, the 1st and 11th.

Solution of the Cases of Oblique-Angled Spherical

Triangles.

PROBLEM. In any oblique-angled spherical triangle, of the three sides and three angles, any three being given, the other three may be found.

In this table, the references (c. 4), (c. 5), &c. are to the cases in the preceding tables.

GIVEN.

SOUGHT.

SOLUTION.

Two sides AB, AC, and the included angle

Let fall the perpendicular CD 1.

from the unknown angle
not required, on AB.

R:cos A=tan AC:tan AD
One of the (c. 2); therefore BD is
other angles, | known, and sin BD:sin
B. AD::tan A : tan B (10);

B and A are of the same
or different affection, ac-
cording as AB is greater or

less than AD (Sp. Ge. 16).

Let fall the perpendicular CD 2. from one of the unknown

angles on the side AB.

R:cos A=tan AC :tan AD The third (c. 2); therefore AD is

side known, and cos AD:cos BC. BD::cos AC:cos BC (9);

according as the segments AD and DB are of the same or different affection, AC and CB will be of the same or different affection.

[ocr errors]
[blocks in formation]

Two angles A and ACB,

and AC, the side between them.

4.

The third

angle

Let fall the perpendicular CD

from one of the given
angles on the opposite side

AB.
R : cos AC : : tan A : cot

ACD (c. 3); therefore the
angle BCD is given, and
sin ACD : sin BCD :: COS
A :cos B (8); B and A
are of the same or different
affection, according as CD
falls within or without the
triangle, that is, according
as ACB is greater or less
than ACD Sp. Ge. 16).

B.

« ForrigeFortsett »