NOTE. -It is shown in geometry that a spherical triangle may, in general, be constructed when any three of its six parts are given (not excepting the case in which the given parts are the three angles). In spherical trigonometry we investigate the methods by which the unknown parts of a spherical triangle may be computed from the above data.

EXAMPLES.

1. In the spherical triangle whose angles are A, B, C,

A+B < §π (1), and A – B <1⁄2 (2).

3. The angles of a triangle are A, 45°, and 120°; find the maximum value of A.

Ans. A <105°.

4. The angles of a triangle are A, 30°, and 150°; find the maximum value of A. Ans. A <60°.

5. The angles of a triangle are A, 20°, and 110°; find the maximum value of A. Ans. A 90°.

6. Any side of a triangle is greater than the difference between the other two.

RIGHT SPHERICAL TRIANGLES.

185. Formulæ for Right Triangles. Let ABC be a spherical triangle in which C is

a right angle, and let O be the centre of the sphere; then will OA, OB, OC be radii: let a, b, c denote the sides of the triangle O opposite the angles A, B, C, respectively; then a, b, and c are the measures of the angles BOC, COA, and AOB.

E

B

α

013

From any point D in OA draw DE L to OC, and from E draw EF 1 to OB, and join DF. Then DE is to EF (Geom. Art. 537). Hence (Geom. Art. 507), DE

Now

:༣o

DF is L to Of

<BC. (Art. 183)

that is, sin b = sin B sin c.

Interchanging a's and b's,

EF EF DF

OF DF OF

sin a sin A sin c.

that is, tan a= cos B tan c.

Interchanging a's and b's,

DE DE EF

OE EF OE

tan b =cos A tan c.

;; that is, tan b = tan B sin a.

Interchanging a's and b's, tan a tan A sin b.

Multiply (6) and (7) together, and we get

Multiply crosswise (3) and (4), and we get

sin a cos B tan c = tan a sin A sin c.

by (1)

Sch. By these ten formulæ, every case of right triangles can be solved; for every one of these ten formulæ is a distinct combination, involving three out of the five quantities, a, b, c, A, B, and there can be but ten combinations in all. Hence, any two of the five quantities being given and a third required, that third quantity may be determined by some one of the above ten formulæ.

186. Napier's Rules.

The ten preceding formulæ, which may be found difficult to remember, have been included under two simple rules, called after their inventor, Napier's Rules of the Circular Parts.

co.c

co. A

Let ABC be a right spherical triangle. Omit the right angle C. Then the two sides a and b, which include the right angle, the complement of the hypotenuse c, and the complements of the oblique angles A and B, are called the circular parts of the triangle. Thus, there are five circular parts, arranged in the figure in the following order: a, b, co. A, co. c, co. B.

co. B

a

Any one of these five parts may be selected and called the middle part; then the two parts next to it are called adjacent parts, and the remaining two parts are called opposite parts. Thus, if co. A is selected as the middle part, then b and co. c are the adjacent parts, and a and co. B are the opposite parts.

Then Napier's Rules are:

(1) The sine of the middle part equals the product of the tangents of the adjacent parts.

(2) The sine of the middle part equals the product of the cosines of the opposite parts.

NOTE 1.- It will assist the student in remembering these rules to notice the occurrence of the vowel i in sine and middle, of the vowel a in tangent and adjacent, and of the vowel o in cosine and opposite.

Napier's Rules* may be made evident by taking in detail each of the five parts as middle part, and comparing the equations thus found with the formulæ of Art. 185.

Thus, let co. c be the middle part. The rules give

sin(co. c) tan (co. A) tan (co. B); .. cos c = cot A cot B

* While some find these rules to be useful aids to the memory, their utility.

others question

...cos B 5= cos b sin A.

NOTE 2.-In applying these rules it is not necessary to use the notation co.c, co. A, co. B, since we may write at once cos c for sin (co. c), etc.

187. The Species of the Parts. — If two parts of a spherical triangle are either both less than 90° or both greater than 90°, they are said to be of the same species. But if one part is less than 90° and the other part is greater than 90°, they are of different species.

In order to determine whether the required parts are less or greater than 90°, it will be necessary carefully to observe their algebraic signs. If the required part is determined by means of its cosine, tangent, or cotangent, the algebraic sign of the result will show whether it is less or greater than 90°. But when a required part is found in terms of its sine, it will be ambiguous, since the sines are positive in both the first and second quadrants. This ambiguity, however, may generally be removed by either of the following principles:

(1) In a right spherical triangle, either of the sides containing the right triangle is of the same species as the opposite angle.

(2) The three sides of a right spherical triangle (omitting bi-rectangular or tri-rectangular triangles) are either all acute, or else one is acute and the other two obtuse.

The first follows from the equation

cos A= cos a sin B,

in which, since sin B is always positive (B<180°), cos A and cos a must have the same sign; i.e., A and a must be either both <or both > 90°.

The second follows from the equation

cos c = cos a cos b.

188. Ambiguous Solution. When the given parts of a right triangle are a side and its opposite angle, the triangle cannot be determined.

For two right spherical triangles ABC, A'BC, right angled at C, may always be

found, having the angles A

and A' equal, and BC, the side opposite these angles, the same in both triangles,

B

but the remaining sides, AB, AC, and the remaining angle ABC of the one triangle are the supplements of the remaining sides A'B, A'C, and the remaining angle A'BC of the other triangle. It is therefore ambiguous whether ABC or A'BC be the triangle required.

This ambiguity will also be found to exist, if it be attempted to determine the triangle by the equation

sin b = tan a cot A,

since it cannot be determined from this equation whether the side AC is to be taken or its supplement A'C.

189. Quadrantal Triangles. The polar triangle of a right triangle has one side a quadrant, and is therefore a quadrantal triangle (Art. 184). The formulæ for quadrantal triangles may be obtained by applying the ten formulæ of Art. 185 to the polar triangle. They are as follows, c being the quadrantal side: