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1. The segment AD is acute or obtuse, according as the ZA is of the same, or of different species with AC. Dc is of the same species with the Z.A. DB is acute or obtuse, according as Bc is of the same, or of different species, with Dc. 2. If DB be less than AD, and their sum less than 180°, then AB is ambiguous; but if their sum be not less than 180°, their difference is the true value of AB, not ambiguous. If DB be not less than AD, and at the same time their sum be less than 180°, this sum is the true value of AB, not ambi

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The z A=51°.80'

Given & The , , Z. B=59° 16' X. Required the side Ac.
The side BC-63°.50%
DETERMINATION OF THE SPECIES.

A perpendicular in this case c

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of the same species, with respect to 180°, Ac is ambiguous; but, if only two of these sums be of the same kind, that value of Ac must universally be taken which agrees with the sum of the angles; in all such cases Ac is not ambiguous.

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BY FORMULA II. page 201. Log sine b-(log sine B+log sine a)—log sine A=80°.19. This example is ambiguous, vide Table II. page 207. EXAMPLE II. The Z. A = 126°.37' Given {#: B = 48°.30 - Required the side Ac.

The side BC = 115°.20%

Answer. Ac-57°.30 acute, and not ambiguous.

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(C) CASE V. Given two angles, and a side opposite to one of them, to find the side adjacent to these angles.

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DETERMINATION OF THE SPECIES. 1. Dc is of the same species as the Z.A. The species of AD cannot be determined. DB is acute, or obtuse,according as Bc and Dc are of the same, or of different species. 2. When the angles A and B are of the same species; if DB+ (AD acute), and DB+(AD obtuse), be each less than 180°, AB is ambiguous; but if only one of these sums be less than 180°, that sum is the true value of AB, not ambiguous. When the angles A and B are of different species; if both AD acute, and AD obtuse, be greater than DB, then AB is ambiguous; but if only one of them be greater, that value diminished by DB leaves AB not ambiguous.

SOLUTION. 1. In the triangle BDC, find the segment DB. #

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mula, vide Table II. page 207.
EXAMPLE II.
The Z A- 126°.37%
Given & The Z. B= 48°.30' X-Required the side AB.
The side BC = 115°.20%

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(D) CASE VI. Given two angles, and a side opposite to one of them, to find the third angle.

The Z A-51°.30'
Given 3. The . . Z. B=59°.16'X-Required the angle c.
The side BC=63°.50'
DETERMINATION OF THE SPECIES.

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tuse, according as the side BC is of the same, or of different species with Dc. 2. When the angles A and B are of the same species; if BCD +(ACD acute), and BCD+(Acd obtuse), be each less than 180°, ACB is ambiguous ; but if only one of these sums be less than 180°, that sum is the true value of ACB, not ambiguous. When the angles A and B are of different species; if both AcD acute, and ACD obtuse, be greater than BCD, then ACB is ambiguous; but if only one of them be greater, that value diminished by BCD leaves ACB, not ambiguous.

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(E) Case VII. Given two sides, and an angle contained between them, to find an opposite angle.

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Determination of THE SPECIES.

1. ad is acute, or obtuse, according as Ac is of the same, or of different species with the Z.A.

B 2. The Z B is of the same, or of different species with the ZA, according as AD is less, or greater than AB.

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(F) Cass VIII. Gives two sides, and the angle contained & Nees tiew, to find the third side. The side Ac- so Given R. The side As-loo.47 X-Required the side BC. The • A- 5ls.So

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