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Fourth Method : (tạn { A) =

– (s—b)(s—c),

8(8-a) log TanĮ A={{log (sb) +log (s-c) -- log s—log (s—a)} + log r.

If A nearly=90°, the first method is inapplicable.

The second method is preferable if A < 90°; the third if A > 90°. If A nearly

180°, the fourth method is inapplicable. (W. Ch. v; L. 72-84; Leg. 53–7; C. 559_-615.)

SPHERICAL TRIGONOMETRY.

(21.) General Principles.

Every section of a sphere made by a plane is a Circle.

The distance between the poles of two great circles is equal to the inclination of their planes.

The pole of a great circle is the pole of all parallel small circles. If the angle subtended by the arc a is invariable, then

a o sin dist. from pole. If the intersections of three great circles are the poles of three others, the intersections of the latter will be the poles of the former.

The sum of the three sides is always < 360°.
The sum of the three angles is always > 180°, and < 540°.
The sum of any two sides is greater than the third side.

The angles at the base of an isosceles spherical triangle are equal; and conversely.

The greater side subtends the greater angle; and conversely.

(W. Ch. viii. prop. l-12; L. 102-66; C. 960_-1010.)

(22.) Relations between the sides and angles of spherical
triangles.
Let S=}(A+B+C),

s=;(a+b+c); and
N= - cos S.cos (S-A).cos (S B).cos (S–C));
n= sin 8.sin (s—a).sin (s—b). sin

12.
c)

S

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S

sin b.sin c.(sin A)=sin (s—b). sin (s -c).
sin b.sin c. (cos; 4) = sin s.sin (s—a).
sin A.sin b.sin c=2n.
sin (a + b) (sin C) = cos(A + B).cos}(A– B). sin c.
sin (a - b)(cos C)=sin}(A + B).sin}(A - B). sin c.

cosc
sin (A + B) = cos (a - b).

cosc

cos C sin}(A - B)= sin (a - b).

sin sc

sinc cos}(A + B) = cos; (a + b).

cos

sinc cos}(A– B)= -sinį (a+b).

sinc sin B.sin C.(sinja)= -cos S.cos (S – A). sin B.sin C.(cosza)=cos (S-B).cos (S–C). sin B.sin C.sin a=2N. sin (A + B) (cos}c)' = cos }(a+b).cos(a - b). sin C.

P

sin (A - B) (sin șc)' = sin}(a + b). sin}(a - b). sin C.

sinc sin(a+b)= cos(A - B).

sinc

sino sin(ab)

sin}(A B). sin C

cos c cos; (a + b)=

sin c cos}(A + B).

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cosc cos(a - b)= sin (A + B).

cos-C sin (a - b)

==(tan_C)?.tan; (A + B). tan}(A - B). sin (a + b) sin (A - B) sin (A + B) = (cot șc)". tanţ(a + b).tanţ(ab). sin (ab).cot c sin (A - B).cot;C

cos (a+b) cos }(A + B) sin A.sin B - sin a.sin b=cos A.cos B.cosc + cos a.cos b.cos C. (sin C). sin (a + b). sin (a - b)=(sin c). sin (A + B).sin (A - B). cos c=cos (a - b)(cosC) + cos (a + b)(sin;C). (sin c)=sin(ab).cosc + sin(a+b).sin C). (cos;c)' =cos (a - b).cos C + cos; (a + b). sin C)*.

sin (a - b) cos B + cos A (tanC)®=

sin (a + b) cos B - cos A cos C= -cos (A B) (sinc) - cos (A + B)(cos c)”. (cos}C)*=sin(A– B). sin c] + sin(A + B).cos c). (sin;C)=cos (A B).sin 5c7 + cos (A + B).cosŚc).

sin (A - B) cos b + cos a (cot c)?

sin (A + B) cos b -- cos a

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cos 6

COS C

=(N.tanja.cotzb.cot;c)).

1 + cos a sin (S-A=

4 cosza.sin b.sinc

sin (s – a). sin (s-6). sin (s—c) sinA.sin, B.sin C=

sin a.sin b.sin c

;

sin s.sin a.sin b.sin c

n.sin s cos z A.cos B.cos?C=

sin a.sin b.sin c

n

tanA.tan B.tan C =

(sin s) cos ža.cosğb.cos že= cos (S – A). cos (S B).cos (S-C)

sin A. sin B.sin C

N
cos S.sin A.sin B.sin C.

Ncos S sinja.sin b.sin c=

sin A.sin B.sin C

N cotža.cotz.b.cot]c=

(cos S):

n=}. (sin a.sin b.sin c)”.sin A.sin B.sin C)*. N=.sin a.sin b.sin c.(sin A. sin B.sin C)2}+. sin s=2..cosA.cosșB.cosșC.

N cos S=2-sin_a.sin b.sin sc.

1

n

COS 8 =

;

COS A

cos B

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1- (sin? A) – (sin; B)– (sino C)

2 sinA.sinB.sin C 1

4 sinA.sinB.sinc

1-(cosža)2 – (çosb)? – (coszc)? sin S=

2 cos a.cosib.cos c
1 +.cos a + cos b + cos C
4 cos a.cosb.cosc

(L. 181-201, tab. VIII.)

n

(23.) Formula for the area (E) of a spherical triangle.

S E

90° = 2S – 7, the radius being the linear unit; =57°,2957795 (2S-180°), the value in square degrees. sin E

2 cosa.coszb.cos
cosE=
cos ža.cos } b + sinja.sinb.cos C

cosc
cos a + cos b + cos C+1

4 cos a.cos.cossic
(cosa) + (costb) + (cosc)? – 1

2 cosa.cosb.cosc
tan E= tans.tanį (s – a). tan] (sb). tan}(s—c).

;

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