The elements of plane trigonometry

Also, Eucl. xI. 21., the three angles forming the solid angle at O are less than four right angles;

arc AB+ arc BC+ arc CA <2π. AO, which is

the circumference of a circle whose radius is AO.

COR. 1. Since the sum of the plane angles which contain any solid angle is less than four right angles, Eucl. x1. 21., it follows from the same mode of proof, that the sum of the sides of any polygonal figure described on a sphere, whose sides are arcs of great circles, is less than the circumference of a great circle.

The polygonal figure, however, must be such that its area is contained on the surface of a hemisphere. For in the proof of the proposition of Euclid to which we have referred, it is supposed that all the plane faces which contain the solid angle may be cut by one plane,-which cannot be the case unless all the edges of the solid angle lie within the same hemisphere.

COR. 2. Also let ABCDE be a five-sided figure described on a sphere, and let it be divided into triangles by the arcs AC, AD. (The Student will easily draw the figure.)

Then AB+BÇ> AC;

AB+ BC + CD > AC + CD, and AC+ CD> AD;

AB+ BC + CD> AD;

AB+ BC + CD + DE>AD + DE> AE.

And the same method of proof being applicable to a polygon of any number of sides, it follows that the sum of all the sides but one of a spherical polygon is greater than the remaining side.

COR. 3. If a and b be two sides of a spherical triangle, since each is less than π,

Also, since any angle of a triangle is less than π;

COR 4.

21.

A-B<T, .. } (4 – B)<¦ .

a', b', c', being the sides of the polar triangle,

a' + b' > c';

· ́`· (π − A) + (π − B) > (π − C) ;

π> A + B - C.

Or, A+ B-C <π.

The sum of the angles of a spherical triangle is greater than two, and less than six, right angles.

Let A, B, C be the angles, and a, b, c the sides of a triangle; A', B', C' and a', b', c', the angles and sides of its polar triangle.

Now 2a + b' + c'.

Art. 20.

> (π − A) + (π − B) + (π − С)

>Sπ- (A+B+ C);

A+B+C>T.

Again, since each of the angles A, B, C is less than π,

22. The angles at the base of an isosceles triangle are equal to each other. (Fig. Art. 28.)

Suppose ABC to be an isosceles triangle, having AB = AC, and therefore ▲ AOB = L AOC.

From D, any point in OA, draw DG perpendicular to the plane BOC, and therefore at right angles to every line it meets in that plane; and from G draw GE and GF perpendicular to OB and OC; join ED, FD, OG.

Then OE OG2 - GE2

(OD2 – DG2) – (ED2 – DG2)

= OD-ED2;

.. DE is perpendicular to OE, and .'. L DEG inclination of the planes BOC and BOA, = ▲ B.

Similarly DF is perpendicular to OC, and DFG = ≤ C.

Now DE: = OD. sin A0B = OD . sin AOC = DF.

And EG2 DE2 – DG2 = DF2 – DG2 = FG2.

Hence, since GE, ED = GF, FD, and GD is common,

.*. ▲ DEG = L DFG, or 4B = C.

23. Conversely, if BC, or 4 DEG = DFG, it may be shewn from the same figure that AB = AC;—that is, the angles at the base being equal, the sides opposite to them are equal.

COR. Hence also it follows that every equilateral triangle is also equiangular; and conversely, that every equiangular triangle is also equilateral.

24. Of the two sides which are opposite to two unequal angles in a triangle, that is the greater which is opposite to the angle which is the greater.

25. Conversely, it may easily be shewn that of two angles in a triangle opposite to unequal sides, that is the greater which is opposite to the side which is the greater.

COR. Hence A B and a b have the same sign.

26. Article 24 has been proved after the manner of Euclid 1. 19. The following propositions may be enunciated for spherical triangles, in the terms used for plane triangles, and may be proved in nearly the same words. Euclid 1. Props. 4. 8. 24. 25. 26. &c.

27. Before we proceed to prove certain formula which are useful for the solution of triangles, we will recapitulate some properties of triangles to which we shall have occasion to refer.

Arts. 14.

14.

20.

24.

A side must be less than a semicircle. An angle must be less than two right angles. 3. Any two sides are together greater than the third.

The greater side is opposite to the greater angle; and conversely.

AB and a b are of the same sign.

a + b + c <2 π.

A+B+C>π, and <3π.