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In this Cafe the Learner will obferve, that there are no two oppofite Parts known; the Hypothenufe A C, oppofite the right Angle at B, being unknown, as are the two Angles A and C, oppofite the given Sides: Hence this can't be folved. by the general Rule, Page 212; but may be folved by the following Proportion :

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To the Tangent of the Angle at A, 360.54, 9.875487

Having found the Angle at A oppofite the given Perpendicular, we have the right Angle at B oppofite the Hypothenufe, the other Part required; which may be found as in Prob. 23; thus,

As the Sine of the Angle at A, 36o. 54′,

Is to the Side B C, 35.66,

So is the Radius, or Sine of the Angle at B,

To the Hypothenuse A C, 59.4,

9.778455

1.552181

10.

11.552181 9.778455

1.773726

The Hypothenuse may be measured by the general Rule, Prob. 19. And, fubtracting 36°. 54, the Angle at A, from 90°, it leaves 53°.6′ for the Angle at C, both which may be measured by Prob. 9.

Oblique

Oblique Plain Trigonometry.

The five Cafes of oblique-angled Triangles are folved by the three following Rules.

When the Sides and Angles are oppofite:

Rule 1. As in right-angled Triangles.
As the Sine of any Angle

Is to its oppofite Side,

So is the Sine of any other Angle

To its oppofite Side:

Or,

As any one Side

Is to the Sine of its oppofite Angle,
So is any other Side

To the Sine of its oppofite Angle.

Rule 2. When any two Sides with the Angle between them are given.

As the Sum of any two Sides

Is to their Difference,

So is the Tangent of the Half-Sum of the two oppofite Angles To the Tangent of Half the Difference of thofe two Angles. Which Half-Difference, being adeed to the Half-Sum, gives the greater of the two Angles; and, being fubtracted from the Half-Sum, leaves the lefs of the two unknown Angles.

Rule 3. When the three Sides are given to find the Angles. As the Base of any Triangle

Is to the Sum of the other two Sides,

So is the Difference of those Sides

To the Difference of the two Segments of the Bafe, made by letting fall a Perpendicular to the Bafe from the Angle oppofite to it.

The Learner being now a little ufed to this Sort of Proportions, it will be proper to fhew him how to perform them without fubtracting the firft Number from the Sum of the fecond and third; which is done by fetting down the arithmetical Complement of the firft Term inftead of the Logarithm.

This may be readily done thus: Subtract the first Figure of the Logarithm to the right Hand from 10, and fet down the •Remainder; then fubtract each of the other Figures, Index and all, from 9, fetting down the Remainders, and place a Dot before the Index, as in the Cafe of the Logarithm. Thus the arithmetical Complement (ufually marked Co. Ar.) of the Logarithm 9.778455 is 0.221545; and fo of any other. When the arithmetical Complement of the firft Term is ufed inftead of the Logarithm, add all the three Numbers together, and reject ten out of the Index of their Sum, as in thofe Cafes where the Radius is the first Term.

Prob. 26. In the oblique-angled Triangle ABC, given the Angle A 36.40', the Angle at C 60°. 51', and the Bafe AC 85.6, to find the other two Sides.

Conftruction. Draw the Bafe A C at Pleasure, and fet off from the diagonal Scale, by the general Rule, Prob. 19, the given Length 85.6, from A to C; then, by Prob. 8, make an Angle of 36°40′ at A, A and an Angle of 60°. 51' at C,

.B

C

and the Meeting of thefe two Lines in B completes the Triangle.

When two Angles of any Triangle are given, their Sum, being fubtracted from 180°, leaves the third Angle. Therefore, in this Cafe, 97°.31', being fubtracted from 180°, leaves 82°. 29 for the Angle at B: Hence we have the Angle at B oppofite the given Bafe, and the Angles at A and C oppofite the two required Sides; which may be found by the first Rule, as follows:

As the Sine of the Angle at B, 82°. 29',
Is to the Bafe A C, 85.6,

So is the Sine of the Angle at A, 36°.40',
To the Side B C, 51.55,

Co. Ar. 0.003748

1.932474 9.776090

1.712312

As the Sine of the Angle at B, 82°. 29',

Co. Ar. 0.003748

Is to the Bafe A, C, 85.6,

1.932474

So is the Sine of the Angle at C, 60°. 51′,

To the Side A B, 75.4,

The Sides BC and A B may be measured by the general

9.941187

1.877409

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But in this Cafe, of two Angles and the included Side given, if the included Side is not the Bafe, it will make fome Difference in the Conftruction; therefore, in the Triangle ABC, the Angle at A 36°. 40', the Angle at B 80°. 29, and the Side A B 75.4 given, to find the other two Sides.

Conftruction. Draw the Bafe AC at Pleasure, and on any Point affurned, as A, make an Angle of 36°. 48', by Prob. 8; take 75.4 from the diagonal Scale, by the general Rule, Prob. 19, in the Compaffes, and fet it from A to

B

B; then, by the fame Prob. 8, make an Angle of 82°. 29′ at B, and draw BC 'till it interfects the Base in C, and the Triangle is completed.

Here we have two Angles given, whofe Sum, being fabtracted from 180°, gives 60°. 51' for the Angle at C; by which the unknown Sides may be found by the firft Rule, as follows:

As the Sine of the Angle at C, 60°.51', Co. Ar. 0.058813 Is to the Side A B, 75.4,

So is the Sine of the Angle at A, 36°. 40′,

--

To the Side BC, 51.55,

1.877371 9.776090 1.712274

And,

As the Sine of the Angle at C, 60°. 51', Co. Ar. 0.058813

Is to the Side A B, 75.4,

So is the Sine of the Angle at B, 82°.29',

To the Bafe A C, 85.6,

1.877371

9.996252

1.932436

The Side B C and Bafe AC may be measured by the general Rule, Prob. 19.

Prob. 27. In the oblique-angled Triangle ABC, given the Bafe AC 85.6, the Angle at B 82°. 29', and the Angle C 60°. 51, to find the other two Sides.

B

Conftruction. Draw the Base AC, and from the diagonal Scale fet off 85.6 from A to C; at C, by Prob. 8, make an Angle of 60°. 51′; then the Angle at A, being the Supplement of the Sum of the other two Angles to 180°, will be 36°.40 by the fame Prob. 8, make an Angle of 36°

A

C

at

at A, and draw the Line A B 'till it interfects CB in the Point B, and the Triangle is completed,

Here having the Angle B oppofite the known Side A C, and the Angles A and C oppofite the required Sides, they may be found by the first Rule, as follows:

As the Sine of the Angle at B, 82°. 29', Co. Ar. 0.003748

Is to the Bafe A C, 85.6,

So is the Sine of the Angle at A, 36°. 40′,

To the Side B C, 51.55,

And,

1.932474 9.776090

1.712312

As the Sine of the Angle at B, 82°. 29', Co. Ar. 0.003748

Is to the Base AC, 85.6,

So is the Sine of the Angle at C, 60°. 51',

To the Side A B, 75.4,

1.932474

9.941187

1,877409

The Sides BC and AB may be measured by the general Rule, Prob. 19.

In this Cafe, of two Angles and an oppofite Side given, if the given Side is not the Bafe, the Conftruction will be fomewhat different; therefore, in the oblique-angled Triangle ABC, given the Side BC 51.56, the Angle at A 36°.401 and the Angle at C 60°. 51, to find the other Sides.

Conftruction. Draw the Bafe AC at Pleasure, and on any Point affumed, as C, by Prob. 8, make an Angle of 60°. 51'; by the general Rule, Prob. 19, fet off 51.56 from C to B; at B, by the fame Prob. 8, A make an Angle of 82°. 29', the

B

Supplement of the two given Angles to 180°, and draw BA 'till interfects the Bafe in A, and the Triangle is completed.

Here we have the Angle A oppofite the known Side B C, the Angle C oppofite the Side A B, and the Angle B, the Supplement of the other two to 180°, oppofite the Bafe A C, the other unknown Side; which may be both found, as in the former Example, by the following Proportion :

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