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CHAPTER XVII.

ON THE SOLUTION OF TRIANGLES.

250. The problem known as the Solution of Triangles may be stated thus: When a sufficient number of the parts of a triangle are given, to find the magnitude of each of the other parts.

251. When three parts of a Triangle (one of which must be a side) are given, the other parts can in general be determined.

There are four cases.

I. Given three sides.

[Compare Euc. I. 8.]

II. Given one side and two angles.

[Euc. I. 26.]

[Euc. I. 4.]

III. Given two sides and the angle between them.

IV. Given two sides and the angle opposite one of

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or,

253. In practical work we proceed as follows:

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10 = } { log (8 − b) + log (s — c) — log s — log (s — a)}.

itan Similarly,

B

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· — 10 = ↓ {log (8 — c) + log (s - a) - log s — log (8 —b)} .

L tan

2

· 1 (s −

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convenient as the tan formula, when one of the angles only

2

is to be found. If all the angles are to be found the tangent formula is convenient, because we can find the L tangents of two half angles from the same four logs, viz. log s, log (s − a), log (s—b), log (s — c). To find the L sines of two half angles we require the six logarithms, viz. log (sa), log (8b), log (sc), log a, log b, log c.

Example.

find A and B.

Given a=275.35, b=189.28, c=30147 chains,

Here, s=383'05, s- a=107·70, s-b=19377, s-c=81.58.
Then

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=10+ {log 193·77+log 81.58-log 383 05 - log 107.70} =10+{2.2872865+1·9115837 −2·5832555 – 2·0322157}

=9.7916995

Α 2

whence =31° 45′ 28′5′′; .. A=63° 30′ 57′′.

[from the tables],

Also

L tan

B

=10+ (log 81.58 + log 107.70 -log 383.05- log 193.77}

=9-5366287=L tan 18° 59′ 9.8";

.. B=37° 58′ 20′′"; C=180° - A-B=78° 30′ 43′′.

255. This Case may also be solved by the formula

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But this formula is not adapted for logarithmic calculation, and therefore is seldom used in practice.

It may sometimes be used with advantage, when the given lengths of a, b, c each contain less than three digits.

Example. Find the greatest angle of the triangle whose sides are 13, 14, 15.

Let a=15, b=14, c=13. Then the greatest angle is A.

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(1) If a=352.25, b=513·27, c=482.68 yards, find the angle A,

having given

log 674-10=2.8287243, log 321.85=2.5076535,

log 160-83-2-2063401, log 191-42=2.2819873,

I tan 20° 38′ =9·5758104, Z tan 20° 39′ =9·5761934.

(2) Find the two largest angles of the triangle whose sides are 484, 376, 522 chains, having given that

log 6.918394780, log 3.15-4983106,

log 2.07=3159703, log 1.69=2278867,

I tan 36° 46′ 6′′=9.8734581, Z tan 31° 23′ 9′′=9.7853745.

(3) If a=5238, b=5662, c= -9384 yards, find the angles A and B, having given

log 1.0142=0061236, log 4.904='6905505,

log 4.48=6512780, log 7.58=8796692,

L tan 14° 38′ =9-4168099, Z tan 15° 57′=94560641,

I tan 14° 39′=9-4173265, Ztan 15° 58′-9.4565420.

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(4) If a=

=4090, b=3850, c=3811 yards, find A, having given log 5.8755 7690448, log 3.85='5854607,

log 1.7855-2517599, log 3.811=5810389,

L cos 32o 15'-9.9272306, L cos 32° 16′-9.9271509.

(5) Find the greatest angle in a triangle whose sides are 7 feet, 8 feet, and 9 feet, having given

log 3=4771213, L cos 36° 42′ =9.9040529,

log 1.4146128, diff. for 60′′ =

=

⚫0000942.

(6) Find the smallest angle of the triangle whose sides are

8 feet, 10 feet, and 12 feet, having given that

log 2=30103, L sin 20° 42'-9.5483585, diff. for 60"- ⚫0003342.

(7) If ab: c=4: 5: 6, find C, having given

log 2=3010300, log 3=4771213,

L cos 41° 25′-9.8750142, diff. for 60" 0001115.

=

(8) The sides of a triangle are 2, √6, and 1+√3, find the angles.

(9) The sides of a triangle are 2, √2 and √3–1, find the angles.

Case II.

256. Given one side and two angles, as a, B, C.

[Euc. I. 26; VI. 4.]

First, A 180° - B-C; which determines A.

=

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or,

.. log blog a + log (sin B) + 10 (10 + log sin A).

log b = log a + L sin B – L sin A.

Similarly, log clog a + L sin C-L sin A.

Example. Given that c=1764.3 feet, B=66° 39′, find b.

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From the Tables we find log 1764.3=3·2465724.

I sin 18° 27'-9.5003421, L sin 66° 39′=9·9628904 ;
.. log b=3.2465724+99628904-9.5003421

=3.7091207=log 5118.2;

..b-5118.2 feet.

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