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(6) Given cos A= and cos B=13, prove that cos C=

(7) If sin2 B+ sin2 C=sin2 4, then A=90o.

(8) If sin 2B+sin 2C=sin 24, then either B=90° or C=90o.

(9) If A.: B: C=1 : 2 : 5, then 1+4 cos A. cos B. cos C=0, and a2, 62, c2 are in A.P.

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(11) If D is the middle point of BC, prove that

4AD2=262+2c3 — a2.

(12) Given that a=2 =26, and that A=3B, prove that C=60o. (13) abc (a cos A + b cos B + c cos C) = 8,S2.

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(15) If D, E, F are the middle points of the sides BC, CA, AB, then

4(AD2+BE2+CF2)=3(a2+b2+c2).

(16) If D is the middle point of BC, cot ADB=

b2-c2

4.S

(17) If d, e, f are the perpendiculars from A, B, C on the opposite sides of the triangle, then

a sin A +b sin B+csin C-2 (d cos A+e cos B+fcos C).

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.

II. Given one side and two angles.

[Compare Euc. I. 8.]

[Euc. I. 26.]

[Euc. I. 4.]

III. Given two sides and the angle between them.

IV. them.

Given two sides and the angle opposite one of [Compare Euc. VI. 7.]

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

Similarly,

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10{log (sc) + log (s - a) - log s-log (s-b)}.

2

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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 (8-6), log (8-c). To find the L sines of two half angles we require the six logarithms, viz. log (sa), log (s-b), log (sc), log a, log b, log c.

Example.

find A and B.

Given a 275 35, b=189.28, c=2
=301 47 chains,

Here, s=383 05, s- a= =107·70,s-b=193 77, 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}
[from the tables],

=9.7916995

A

whence -31° 45′ 28.5′′; .. A=63° 30′ 57′′.

Also

B

Ltan 10+ (log 81.58+ log 107-70-log 383.05- log 193.77}

2

=

-9.5366287=Ltan 18° 59′ 9.8";

.. B=37° 58′ 20′′; C=180o – 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) Ifa=352.25, b=513.27, c=482.68 yards, find the angle 4, 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, L 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 4.486512780, log 7.58='8796692,

I tan 14° 38' 9.4168099, L tan 15° 57'-9.4560641,
L tan 14° 39′ = 9·4173265, Ztan 15° 58′ =9-4565420.

(4) If a=4090, b=3850,

C= =3811 yards, find A, having given

log 5.87557690448, log 3.85='5854607,

log 1.7855-2517599, log 3.811=5810389,

L cos 32° 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 1.4=146128, 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 a:b: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.

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