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The above result may be enunciated thus-

The sum of two sides is to their difference as the tangent of half the sun of their opposite angles is to the tangent of half their difference.

By combining the last two formulæ the formula for the tangent of half an angle in terms of the sides is easily deduced, as follows

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(log. (s — b) + log. (s—c) + 20 — (log. s + log. (s –

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2

SPHERICAL TRIGONOMETRY

The fundamental formula in spherical trigonometry as deduced from the spherical triangle having its solid angle at the centre of the sphere is that which connects the cosine of an angle with sines and cosines of the three sides of the triangle.

In the spherical triangle A B C, Fig. 10, given the three sides a b c to find angle A.

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and substituting the reciprocals of sin. b and sin.

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L. sin. s + L. sin. (s — a) + L. cosec. 6. L. cosec. C-20

20}

2

TO FIND LOGARITHMIC FORMULA FOR THE SINE OF AN ANGLE IN

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Let } (a + b + c) =s; then } (a + bc) =s-c; and 1 (a - b + c) = S

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А L. sin.

2

{ {L

{L. sin. (s—c) + L. sin. (s -- b) + L. cosec. 6 + L. cosec. c

20

By combining these two formulæ the tangent formula is easily deduced as follows

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TO PROVE A FORMULA FOR FINDING A DIRECT THROUGH THE

NATURAL HAVERSINE

Cos. a = cos. b.cos. c + sin. b. sin. c — sin. b. sin. c (1 cos. A)
= cos. (6 ~ c) -- sin. b. sin. c (I cos. A)

cos. (6 ~ c) + sin. b. sin. c ( 1 - cos. A)
COS. a
cos. (b c)

(1 cos. A)
+ sin. b. sin. c

I

COS. a = I

I

2

2

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For the further use of the natural ha versine, see the Explanation of the Haversine Tables in NORIE'S TABLES.

The usual formula for finding an angle is as followswhere a the alt. I = the latitude, p = the polar dist., and _P the hour angle.

cos. (90° --- () - cos. p cos. (90° — 1)

sin. p sin. (90° — 1)

cos. p sin. ? sin. p cos. l.

cos. P =

sin. a

sin, a

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i.e.,

sin a

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Subtracting each side of the equation from 1-
I- cos. P

COS. p sin, 1

sin. p cos. l
Р sin. p cos 1 + cos p sin. I - sio. a
2 sin.
2

sin p cos. I
sin. (p + 1)

sin. p cos. I
2 cos. } +1 + a) sin. 1 (+1- a)

sin. p cos. 1 Now, if

s= } (P + 1 + a). then (s - a) = } ( +1-a)

Р cos. s sin. (s-a) therefore

sin.

sin. p cos. 2
= co-sec. p sec. I cos. s sin. (s — a)

P
And

sin. V co-sec. p sec. I cos. s sin. (s—a)
2

P
Hav. P = sin.2

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2

In finding the azimuth when the altitude and latitude are substituted for zenith distance and co-latitude, the following modification of the formula is necessary.

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adding 1 to each side of the equation

I-cos. Z=1+

cos. --sin. a sin. I

cos, a cos. 1
z ícos. a cos. 1-sin. a sin. I) + cos.
... sin.'

cos, a cos. 1

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z

Hav. Z=

sin.2

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