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dicular to OB and OC respectively. Join L and M; join L and N. Then LMOB, and LN1 OC.

(Geom.)

Now,

Angle B is measured by angle LMG.
Angle C is measured by angle LNG.

M

cos a'

cos b'

cos c'

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LM. sin B = LG: = LN sin C;
LM =OL sin AOB=OL sin c;
LNOL sin AOCOL sin b;

.. OL sin C sin B=OL sin b sin c.

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N

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B

FIG. 66.

182. If A'B'C' be the polar triangle of ABC, then (Fig. 67)

a' = 180° A;

b' = 180° - B;

c' 180° C.

=

=

We may now apply formula (1) to A'B'C'.

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

cos a'

=

cos A, sin b'
cos B, sin c'
cos C, cos A'= == cos a.

-

a

C

=

sin B;

sin C';

(Art. 177.)

(a)

B'

cos A = cos B cos C

Substituting these values in (a), we obtain

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183. From (1),

cos Acos B cos C-sin B sin C cos a.

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Treating the other two formulæ similarly, we obtain

cos B cos C+ sin B sin C cos a;
cos C cos A+ sin C sin A cos b;
cos A cos B+ sin A sin B cos c.

a

a' FIG. 67.

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Α

cos b =

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COS C =

Β

Relation involving the angles and one side.

Solving these equations for cos a, cos b, cos c, we have

cos A+ cos B cos C
;
sin B sin C
cos B+ cos C cos A
sin C sin A

;

cos C+ cos A cos B
sin A sin B

(4)

(5)

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

Substituting in this formula the value of cos c obtained from (1), we obtain

or

cos a (1 — cos2 b) = sin a sin b cos b cos C + sin b sin c cos A; cos a sin b = sin a cos b cos C + sin c cos A; and similarly, cos b sin a = sin b cos a cos C+ sin c cos B; cos b sin c = sin b cos c cos A+ sin a cos B; cos e sin b = sin c cos b cos A+ sin a cos C; cos e sin a sin c cos a cos B+ sin b cos C ; cos a sin c = sin a cos c cos B+ sin b cos A.

(6)

(6) is a relation involving two angles and the sides. By treating (3) similarly,

cos A sin B:
= cos a sin C- cos c cos B sin A;
cos C sin B = cos c sin A-cos a cos B sin C;
cos C sin A: = cos c sin B-cos b cos A sin C;
cos B sin A
cos B sin C
cos A sin C

= cos b sin C cos c cos A sin B;
= cos b sin A - cos a cos C sin B;
= cos a sin B-cos b cos C sin A.

(7) is a relation between the angles and two sides.

From (6),

cos a sin b = sin a cos b cos C+ sin c cos A;
cos A;

sin b = cos b cos C+

sin c
sin a

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

cos A
sin A

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

The student may derive the five corresponding formulæ :

...

From (2),

similarly,

184. To express cos▲, sin A, tan ▲ in terms of the sides. 1/12/14

(Art. 92.)

cos A=

cos 4+1
2

cos A

cos B

=

cos C

=

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cos A+ 1
2

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sin s

sin b sin c

If we put a + b + c = 2 s, («) may be written:

sin (s a) sin b. sinc

sins sin(sb)

sin a sin c

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2 sin b sin c

cos (b+c)

sin

cos b cos c

sin s. sin(sc),

sin a sin b

(b+c-a

(b+

2

(7)

(Art. 180.)

(8)

(Art. 75.)

(a)(Art. 82.)

(9)

EXERCISE. The student may show that

... sina

sinA =

tan A =

cosa =

K

with the corresponding formulæ for B and C.

sina =√

2
sin B sin C

185. To find cosa, sina, tana. Using the results of (5), the student may show that, if S

=

(A+B+C),

=

sin a =

sin(sb) sin(s
sin b sin c

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cos (S

1

sin(s - b) (s - c)

sins sin(sa)

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cos a

tana = cos(S

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cos S cos (S

A)

(13)

(14)

(Art. 91.) cos S cos (S — A) cos (S – B) cos (S – C). (15)

A)

B) cos (S-C)

2 sina cosa

(10)

(11)

(12)

186. I. Since any side a < 180°, and any angle A< 180°, cosa, sina, tana, sin A, cos 1⁄2 A, tan ↓ A

are all positive; hence the sign of the radical in expressions (9)—(15) is positive.

II. Since s<180°, a < 180°, b < 180°, c<180°, and since the differences s a, sb, sc are less than 180° and positive, the expressions (9)-(11) are real.

III. If a', b', c' are the sides of the polar triangle of A, B, C, then a'<(b'+c');

i.e.

180° A<(180° B+ 180° — C'); B+C-A<180°;

i.e.

i.e.

S - A < 90°;

.. cos (S — A) is positive.

Since 270° > S > 180°, cos S is negative; i.e. - cos S is positive. Therefore the expressions (12) - (15) are real.

187. cos(A+B)= cos A cos B- sin A sin Substitute in this equation the values of cos and (10).

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sin

c)

A B

2

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=

2

B

sin (sc) sin (s - a) sin (sc)
sin c
sin b sin a

28 C

2

2 sine cos c

2 cos

COS

sin (sa) (sc)

sin b sin a

COS

sin

a+b
2

cos c

sin (s-a) sin (s—b)

sin a sin b

a+b

2

COS

a+b

2

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C sin 2

sin

(a - b) sin

2

====

sin

COS

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B. (Art. 75.)

A, etc., from (9)

sin

23

COS

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(from (10)).

C

2

Formulæ (16), (17), (18), (19) are known as Gauss' Formulæ.

(16)

(17)

(18)

(19)

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