Treatise on Geometry and Trigonometry

CHAPTER XIII.

SPHERICAL TRIGONOMETRY.

877. SPHERICAL TRIGONOMETRY is the investigation of the relations which exist between the sides and angles of spherical triangles.

Each side of a spherical triangle being an arc, is the measure of an angle. It has the same ratio to the whole circumference that its angle has to four right angles. It may be measured by degrees, minutes, and seconds, as an angle is measured. It has its sine, tangent, and other trigonometrical functions; it being understood that the sine, etc., of an arc are the sine, etc., of the angle at the center which that arc subtends.

The propositions which express the relations between the sides and angles of a spherical triangle, apply equally well to the faces and diedral angles of a triedral (766) and seq.). If the investigation were made from this point of view, as it well might be, the proper title of the subject would be Trigonometry in Space.

THREE SIDES AND AN ANGLE.

878. Theorem.-The cosine of any side of a spherical triangle is equal to the product of the cosines of the other two sides, increased by the product of the sines of those sides and the cosine of their included angle.

Let ABC be a spherical triangle, O the center of the

sphere, AD and
AE tangents re-
spectively to the
arcs AB and AC.
Thus, the angle

EAD is the angle
A of the spherical O

triangle; the angle

EOD is measured

by the side a, and

so on.

From the triangles EOD and EAD (865),

DE2=OD2+ OE-20D OE cos. a,

DE2= = AD2+AE2—2AD·AE cos. A.

By subtraction, the triangles OAE and OAD being right angled,

0 =

20A+2AD AE cos. A-20D.OE cos. a;

that is,

cos.acos. b cos. c + sin. b sin. c cos. A.

In the above construction, the sides which contain the angle A are supposed less than quadrants, since the tangents at A meet OB and OC produced. That the formula just demonstrated is true when these sides are not less than quadrants, is shown thus:

Then, in the triangle AB'C, as just demonstrated,

cos. a' cos. b cos. c' + sin. b sin. c' cos. B'AC.

Now, a', c', and B'AC are respectively supplements of a, c, and BAC. Hence,

cos.acos. b cos. c + sin. b sin. c cos. A.

When both the sides which contain the angle A are greater than quadrants, produce them to form the auxiliary triangle, and the demonstration is similar to the last.

Suppose that one of the sides b and c is a quadrant, for example, c. On AC, produced if necessary, take AD equal to a quadrant, and join BD. Now A is a pole of the arc BD (754), and therefore that arc measures the angle A (760).

but

Then, from the triangle BCD,

cos. a=cos. CD cos. BD+ sin. CD sin. BD cos. CDB;

CD is the complement of b, BD measures A,

and CDB is a right angle. Hence, this equation be

comes,

cos. a= sin. b cos. A,

and the formula to be demonstrated reduces to this, when c is a quadrant.

The proposition having been demonstrated for any angle of any spherical triangle,

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

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

These have been called the fundamental equations of Spherical Trigonometry. By their aid, when any three of the elements of a spherical triangle are known, the others may be calculated.

A SIDE AND THE THREE ANGLES.

879. Since the formulas just demonstrated are true of all spherical triangles, they apply to the polar triangle of any given triangle. Therefore, denoting the sides and angles of the polar triangle, by accenting the letters of their corresponding parts in the given triangle,

cos. a' cos. b' cos. c' + sin. b' sin. c' cos. A',