In the right triangle ABC in which the angle C is the right angle, prove the following relations:

1. sin2 asin2 b - sin2 c = sin2 a sin2 b.

10. If a = b = c, prove sec A = 1 + sec a.

11. If c< 90°, show that a and b are of the same species. 12. If c> 90°, a and b are of different species.

13. A side and the hypotenuse are of the same or oppo

π

site species, according as the included angle <, or > 2

OBLIQUE SPHERICAL TRIANGLES.

190. Law of Sines. In any spherical triangle the sines of the sides are proportional to the sines of the opposite angles. Let ABC be a spherical triangle, O the centre of the sphere; and let a, b, c denote the sides of the triangle opposite the angles A, B, C, respectively. Then a, b, and c are the measures of the angles BOC, COA, and AOB.

From any point D in OA draw DGL to the plane BOC, and from G draw GE, GF 1 to OB, OC. Join DE, DF, and GO. Then DG

E

B

C

is to GE, GF, and GO (Geom. Art. 487). Hence, DE is 1 to OB, and DFL to OC (Geom. Art. 507).

or

≤ ≤ .. DEGB, and DFGZC.. (Art. 183)

In the right plane triangles DGE, DGF, ODE, ODF, DG = DE sin B = OD sin DOE sin B = OD sin c sin B, DGDF sin C = OD sin DOF sin C = OD sin b sin C. sin c sin B: = sin b sin C ;

...

sin b: sin c :: sin B: sin C.

Similarly, it may be shown that

sin a sin c: sin A: sin C.

NOTE. The common value of these three ratios is called the modulus of the spherical triangle.

Sch. In the figure, B, C, b, c are each less than a right angle; but it will be found on examination that the proof will hold when the figure is modified to meet any case which can occur. For example, if B alone is greater than

90°, the point G will fall outside of OB instead of between OB and OC. Then DEG will be the supplement of B, and thus we shall still have sin DEG = sin B.

191. Law of Cosines. In any spherical triangle, the cosine of each side is equal to the product of the cosines of the other two sides, plus the product of the sines of those sides into the cosine of their included angle.

Let ABC be a spherical triangle, O the centre of the sphere, and a, b, c the sides of

the triangle opposite the angles A, B, C, respectively. Then

a = ≤ BOC,

b=/ COA,

c = ZAOB.

From any point D in OA draw, in the planes AOB, AOC, respectively, the lines DE, DFL to OA.

Then

(Art. 183)

Join EF; then in the plane triangles EOF, EDF, we

have

EF2 = OE2 + OF2 - 2 OE. OF cos EOF

EF2 = DE2 + DF2 - 2 DE DF cos EDF .

also in the right triangles EOD, FOD, we have

.

OE2 – DE2 = OD2, and OF2 - DF2=OD2. (3)

Subtracting (2) from (1), and reducing by (3), and transposing, we get

20E. OF cos EOF = 2 OD2 + 2 DE. DF cos EDF.

or

OD OD DF DE

.. cos EOF =

+

cos EDF,

OF OE OF OE

cos a = cos b cos c + sin b sin c cos A (4)

By treating the other edges in order in the same way, or by advancing letters (see Note, Art. 96) we get

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

(5)

(6)

Sch. Formula (4) has been proved only for the case in which the sides b and c are less than quadrants; but it may be shown to be true when these sides are not less than quadrants, as follows:

(1) Suppose c is greater than 90°. Produce BA, BC to meet in B', and put AB'c', CB'= a'.

C

a

B

Then, from the triangle AB'C, we have by (4)

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

>B'

or cos (π-α) = cos b cos (π−c) + sin b sin (π — c) cos (π —A).

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

Then, from the triangle A'BC, we have by (4)

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

The triangle AB'C is called the colunar triangle of ABC.

192. Relation between a Side and the Three Angles. In any spherical triangle ABC,

cos A

-=

cos B cos C+ sin B sin C cos a.

Let A'B'C' be the polar triangle of ABC, and denote its angles and sides by A', B', C', a', b', c'; then we have by (4) of Art. 191

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

cos C

Rem.

This process is called "applying the formula to the polar triangle." By means of the polar triangle, any formula of a spherical triangle may be immediately transformed into another, in which angles take the place of sides, and sides of angles.

193. To show that in a spherical triangle ABC,

cot a sin b = cot A sin C + cos C cos b.

Multiply (6) of Art. 191 by cos b, and substitute the result in (4) of Art. 191, and we get

cos a = cos a cos2 b + sin a sin b cos b cos C + sin b sin c cos A. Transpose cos a cos2 b, and divide by sin a sin b; thus,

= cos b cos C + cot A sin C. (by Art. 190)

By interchanging the letters, we obtain five other formulæ like the preceding one. The six formulæ are as follows: