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CASE S VI. and VII.

The Legs given, to find the Angles and Hypothenufe.

The Legs A B 980, B C 690 given, to find the Angle B A C, og ACB, and the Hypothenuse A C.

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To find Angle C.

As the Base 690

2.83885 As Radius

Is to Radius

10.00000 Is to the Base 690

So is the Perpend. 980

By making the Base Radius, it will be,

To find the Hypoth. A C.

10.00000

2.83885

2.99123 So is Sec. Ang. C 54° 51' 10.23979

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To Tang. Ang. C 54° 51′ 10.15238 To the Hypoth. 1198

By making the Perpendicular Radius, it will be,

3.07864

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9′

13.07866

10,00000

To Tan. Ang. A 35° 9' 9.84762 To the Hypoth: 1198

By GUNTER.

3.07866

"The Extent from 690 to 980 on the Line of Numbers, will reach from Radius (or 45 Degrees) to 54.51 on the Line of Tangents.'

2dly. The Extent from Sine Angle A 35 Degrees 9 Minutes to Radius, or 90 Degrees, will reach from the Bafe 690 to the Hypothenufe 1198, on the Line of Numbers.'

NOTE. When the Degrees and Minutes of one Angle of a RightAngled Triangle is found, the other Angle is alfo given in the Tables, without fubtracting from 90°, whether in Sines, Tangents, or Secants, for one Angle is always the Complement of the other; therefore, if the Degrees at the Top of the Tables and the Minutes in the Left Hand Column, which always correfpond with the Logarithmic Sine, Tangent or Secant increafing downwards, are found; then the Degrees less than 90°, belonging to the other Angle, will be at the Bottom of the Tables, and the Minutes in the Right Hand Column; and, on the contrary, if the Degrees at the Bottom, and the Minutes in the Right Hand Column, correfpond with the Logarithmic Sine, Tang. &c. then the Degrees of the other are at the Top, and the Minutes in the Left Hand Column.

Questions to exercise the Learner in Trigonometry.

Queft. 1. The Hypothenufe 98 Miles and the Angle oppofite to the Bafe 33° 45′ given, to find the Bafe and Perpendicular. Anf. Bafe 54.45, and the Perpend. 81.48 Miles.

Quest. 2. The Perpend. 82 Leagues, and the Angle opposite to the Bafe 33° 45' given, to find the Hypoth. and Bafe.

Anf. The Hypoth. 98.62, and Bafe 54.79 Leagues.

Queft. 3. The Bafe 82 Yards, and the Angle oppofite to the Perpend. 33° 45' given, to find the Hypoth. and Perpendicular. Anf. Hypoth. 98.62, and the Bafe 54.79 Yards.

Quest. 4. The Hypoth. 126, and Perpend. 70 Miles given, to find the Bafe.

Anf. Bafe 104.8 Miles.

Queft. 5. The Hypoth. 124, and the Bafe 86 Miles given, to find the Perpend.

Anf. Perpend. 89.35 Miles.

Queft. 6. The Bafe 64, and Perpend. 114 Leagues given, to find the Hypoth.

Anf. Hypothenufe 130.8 Leagues.

OBLIQUE

OBLIQUE TRIGONOMETRY.

AXIOM II.

IN all plane Triangles, the Sides are in direct Proportion to the

Sines of their opposite Angles.*

CASE I.

Two Angles and one Side given, to find either of the other Legs. The Angle B D C ro1° 25' and C BD 44° 42', and the Leg BC 76 given, to find the Sides CD and B D.

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*Let the Triangle A B C be circumfcribed by a Circle, it is plain that each Side becomes a Chord to its refpective Arch; from hence, as the Chord of the Arch A C is to the Chord of the Arch CB, A

fo is the Side A C to the Side C B.

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1.73669

C

B

Now it has been fhewn, that half the Chord is the Sine of the Arch fubtended by that Chord; therefore in the Triangle A B C, the Sines of the Angles will be as the Halves of their oppofite Sides; and fince the Halves are as Wholes, it follows, that the Sines of the Angles are as their oppofite Sides; that is, AC: C B :: Sine Ang. B: to Sine Ang. A, &c.

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Extend from the Supplement of Angle D 78° 35′ to Angle B 44° 42′ on the Line of Sines, that Extent will reach from the Side BC 76, to the Side D C 54,53 on the Line of Numbers.’

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2dly. Extend from the Supplement of Angle D 78° 35' to Angle C33° 53' on the Line of Sines, that Extent will reach from the Side BC 76, to the Side B D 43.23 on the Line of Numbers."

CASE S I. and II.

Two Sides and an Angle opposite to one of them given, to find the other oppo fite Angle and third Side.

The Side B C 106, D B 65 Miles, and the Angle BCD 31° 49′ given, to find the Angle B D C obtufe, and the Side C D.

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Here it may be proper to obferve, that if the given Angle be Obtufe, the Angle fought will be Acute; but when the given Angle is Acute, and oppofite to a leffer given Side, then the required Angle is doubtful, whether Acute or Obtufe; it ought, therefore, to be determined before the Operation. For it is plain the above Proportion produces 59° 17' for the required Angle; but as it is Obtufe, its Supplement to 180 Degrees must be taken, viz. 120° 43"

By GUNTER.

The Extent from 65 to 106 on the Line of Numbers, will reach from 31° 49′ to 59° 17′ on the Line of Sines.'

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2dly. The Extent from 31° 49′ to 27° 28' on the Line of Sines, will reach from 65 to 56.88 on the Line of Numbers.'

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In every Plane Triangle it will be, as the Sum of any two Sides is to their Difference, fo is the Tangent of half the Sum of the Angles oppofite thefe Sides to the Tangent of half their Difference.

Which half Difference, being added to half the Sum of the Angles, gives the greater, but if fubtracted, the Remainder will be the leffer Angle.**

G

* Produce the leffer Leg C B of the Triangle A B C, till B D becomes equal to BĂ; then bifect D C in E; join A Dand bifect it also in F, which (by z. 6. Eucl.) will be perpendicular to A D, and draw E F, which (by 2. 6. Euclid) will be parallel to A C. Then will the Angles A B F F B D A B D, which external Angle ABD is (32. 1. Eucl.) A B C+B C A, that is the Sum of the oppofite Angles required.

BE

Then draw B G parallel to C A, fo will the Angle G B A be (by 29. 1. Eucl.) equal to its alternate one B A C. And if from half the Sum of the

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