SPHERICAL TRIGONOMETRY.

1. A SPHERICAL TRIANGLE is a portion of the surface of a sphere included by the arcs of three great circles (B. IX., D. 1). Hence, every spherical triangle has six parts; three sides and three angles.

2. SPHERICAL TRIGONOMETRY explains the processes of determining, by calculation, the unknown sides and angles of a spherical triangle, when any three of the six parts are given. For these processes, certain formulas are employed which express relations between the six parts of the triangle.

3. Any two parts of a spherical triangle are said to be of the same species when they are both less or both greater than 90°; and they are of different species, when one is less and the other greater than 90°.

4. Let ABC be a spherical triangle, and P the centre of the sphere. The angles of the triangle are

equal to the diedral angles included Pbetween the planes which determine

its sides; viz.: the angle A to the angle included by the planes PAB

B

N

M

a

and PAC; the angle B to the angle included by the planes PBC and PBA; the angle C to the angle included by the planes PCB and PCA (B. IX., D. 1). If we regard the side PA as unity, the sides CB, CA, AB, of the spherical triangle will measure the angles CPB, CPA, APB, at the centre of the sphere. Denote these sides or angles, respectively, by a, b, and c.

5. On PA, the intersection of two faces, assume any point, as M, and in the planes APB, APC, draw MN and

21

MO, both perpendicular to the com
mon intersection PA: then, OMN
will measure the angle between these
planes (B. VI., D. 4), and hence, will P-
be equal to the angle A of the tri-
ngle. Join O and N by the straight
ine ON.

M

In the triangles NPO and NMO, we have (Plane Trig,.

and by reducing to entire terms,

2PNXPOXcos a=PN+PO-NO; 2MOX MNXcos A-MN+MO-NO. By subtracting the second equation from the first, we have

=

PNXPOX cosa - MOX MN cos A) - PN-MN+PO'—MO— 2PM' and by dividing both members by 2PN × PO, we have,

cos a = cos b cos c + sin b sin c cos A.

A similar equation may be deduced for the cosine of either of the other sides: hence,

cos a = cos b cos c + sin b sin c cos A,

cos b = cos a cos c + sin a sin c cos B,

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

}

(1)

That is: The cosine of either side of a spherical triangle is equal to the product of the cosines of the two other sides plus the product of their sines into the cosine of their included angle.

The three equations (1) contain all the six parts of the spherical triangle. If three of the six quantities which

enter into these equations be given or known, the remaining three can be determined (Bourdon, Art. 103): hence, if three parts of a spherical triangle be known, the other three may be determined from them. These are the primary formulas of Spherical Trigonometry. They require to be put under other forms to adapt them to logarithmic computation.

6. Let the angles of the spherical triangle, polar to ABC, be denoted respectively by A', B', C', and the sides by a', b', c'. Then (B. IX., P. 6),

Since equations (1) are equally applicable to the polar triangle, we have,

cos a' cos b' cos c' + sin b' sin c' cos A':

substituting for a', b', c' and A', their values from the polar triangle, we have,

- cos A = cos B cos C - sin B sin C cos a; and changing the signs of the terms, we obtain, cos Asin B sin C cos a cos B cos C

Similar equations may be deduced from the second and third of equations (1); hence,

That is: The cosine of either angle of a spherical triangle, is equal to the product of the sines of the two other angles into the cosine of their included side, minus the product of the cosines of those angles.

7. The first and second of equations (1) give, after transposing the terms,

cos a + cos b -cos e (cos a + cos 6)=sin (sin b cos + sin a cus B);

and by substracting the second from the first,

coe a- cos b + cos c (cos acos b) = sin c (sin b cos Asin a cos B), these equations may be placed under the forms,

(1 cos c) (cos a + cos b) = sin c (sin b cos A + sin a cos B), (1 + cos c) (cos a cos b) = sin c (sin b cos A — sin a cos B); multiplying these equations, member by member, we obtain, (1 - cos2 ) (cos2 a — cos2 b) = sin2c (sin2 b cos2 A— sin2 a cos2 B):

substituting sin c for 1 - cos2 c, 1 cos c, 1 sin2 A for cos2 A, and 1- sin2 B for cos2 B, and dividing by sin2 c, we have,

sin2 a, we have,

then, since cos2 a - cos2 b = sin? b

sin b sin A = sin2 a sin2 B;

and, by extracting the square root,

sin b sin A = sin a sin B.

By employing the first and third of equations (1) we shall find,

sin c sin A = sin a sin C;

and, by employing the second and third,

sin b sin = sin c sin B; hence,

That is: In every spherical triangle, the sines of the angles are to each other as the sines of their opposite sides.

8. Each of the formulas designated (1) involves the three sides of the triangle together with one of the angles. These formulas are used to determine the angles when the three sides are known. It is necessary, however, to put

them under another form to adapt them to logarithmio computation.

Taking the first equation, we have,

But, 1 + cos A = 2 cos2 A (Plane Trig., Art. 85),

and, sin b sin ccos b cos c = cos (b + c) (Art. 73);

} {
s = (a + b + c) and }s — a = § (b + c − a) :

9. Had we subtracted each member of the first equa tion in the last article, from 1, instead of adding, we should, by making similar reductions, have found,