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SOLUTION OF THE CASES OF PLANE TRIANGLÉS.

73. There are three ways of solving trigonometrical problems, viz. by geometrical construction, by arithmetical computation, and instrumentally, or by the scale and compasses. The first of these methods has been already explained in part 8. under the head of Practical Geometry; the second consists in the application of the principles laid down in the foregoing theorems, by the help of either natural numbers, or logarithms; and by the third, the proportions are worked with a pair of compasses on the Gunters' scale; the method of doing which will be explained in the following examples, where the conditions are exhibited in the form of a Rule of Three stating, having either the first and second terms, or the first and third, always of the same kind. 74. When the first and second terms are of the same kind.

Extend the compasses from the first term to the second, on that line of the Gunter which is of the same name with these terms; this extent will reach from the third term to the fourth, on the line which is of the same name with the third and fourth. 5. When the first and third terms are of the same kind.

Extend the compasses (on the proper line) from the first to the third; that extent will reach (on the proper line) from the second to the fourth; observing in all cases, that when the proportion is increasing, the extent must be taken forwards on

This scale was invented by the Rev. Edmund Gunter, B.D. professor of Astronomy at Gresham College, probably about the year 1624; it is a broad flat ruler two feet in length, on which are laid down (besides all the lines common to the plane scale) logarithmic lines of numbers, sines, versed sines, tangents, meridional parts, equal parts, sine rhumbs, and tangent rhumbs; that is, the actual lengths (taken on a scale of equal parts) are expressed by the figures constituting the logarithms of the quantities in question. With these logarithmic scales, all questions relating to proportion in numbers may be solved, for-the compasses being extended from the first term to the second or third, that extent will reach from the second, or from the third to the fourth, according as the first and second, or first and third terms are of the same kind. For an ample description of this scale, see Robertson's Elements of Navigation, vol. 1. p. 114. 4th. edit, likewise Mr. Donne's directions usually sold with his improved scale; and for an account of the improvements by Mr. Robertson, see a tract on the subject, published in 1778, by William Mountaine, Esq. F. R. S.

the scale, but when the proportion is decreasing, it must be taken backwards.

SOLUTION OF RIGHT ANGLED TRIANGLES.

76. Case 1. Given the hypothenuse AB, and one side AC, of a right angled triangle; to find the remaining side BC, and the angles A and B ‘.

2

Because BC+ACAB (47.1.) BC2=AB-AC2,

and BC=AB-AC, whence BC is found. Likewise (Art. 63.) hyp. AB side AC :: radius sin. angle B; that is, sin B= ACX radius

AB

; or by logarithms, log. sin B

=log. AC+log. rad.-log. AB; whence the angle B is found, both by natural numbers and logarithms.

Lastly, since the three angles of any tri

angle are equal to two right angles (32. 1.) B

=180°, and the angle C (a right angle)=90°, ·.· B+A= (180°-C=190°-90°) 90°; but the angle B has been found, · A=90—B is likewise known ".

By a similar process AB and BC being given, AC and the angles B and A may be found.

• Before you begin to work any question in Trigonometry, you must draw a sketch resembling, as nearly as you can guess, the figure intended; placing letters at the angles, and each number given in the question opposite the side or angle to which it belongs; some authors mark the given sides and angles by a small stroke, drawn across the given side, or issuing from the given angle; the unknown parts they mark with a cipher (0).

It must be remembered, that multiplication of natural numbers is performed by the addition of their logarithms, division by subtraction, involution by multiplication, and evolution by division; if these particulars be kept in mind, there will be no difficulty in solving trigonometrical problems by logarithms, see vol. 1. part. 3.

The angle A may be found in the same page of the table in which B is found; thus, if the degrees and minutes contained in B be found at the top and on the left hand respectively, of the page, those contained in A will be found at the bottom and on the right ; viz. the degrees at the bottom of the page, and the minutes on the right hand, in a line with the minutes in B.

EXAMPLES.-1. Given the hypothenuse AB=120, and the perpendicular AC=95, to find the base BC and the angles A and B.

By construction.

Draw any straight line BC, at C draw CA perpendicular to BC, and make it equal to 95 taken from any convenient scale of equal parts; from A as a centre with the radius 120 taken from the same scale, cross CB in B, and join AB. Take the length of CB in the compasses, and apply it to the above. mentioned scale, and it will be found to measure 73 nearly; next measure the angles A and B by the scale of chords or the protractor, and they will be known, viz. A=38° and B=52′, nearly 1.

By calculation.

2

First, to find BC. We have BC=√AB\2—AC{2= (/120-952=5375=) 73.3143, &c.

Secondly, to find the angle B. We have sin B= ACX rad 95 x 1 =).7916666 the natural sine of B, and the

AB

=

120

nearest angle in the table corresponding with this sine is 520 20''; wherefore the angle B-52° 20′, and A=(90°—B=90°— 52° 20′ ) 37° 40'.

The sides and angles of triangles are very expeditiously determined both by the plane scale and the Gunter, but these methods are not to be depended on in cases where accuracy is required; they are nevertheless useful where great exactness is no object, and as convenient checks on the method of calculation.

y The side BC may likewise be found trigonometrically, after the angle A AB.sin A

has been found; thus (Art. 63.) AB : BC : : rad : sin A,·.· BC=" rad

this solution may be performed by the Gunter; thus, extend on the sines from 90° to 37°, this extent will reach on the numbers from 120 to 731-BC nearly.

z This, although it is the angle which has the nearest sine in the table to the above, is not perfectly exact; the natural sine of 52° 20′ being only 7915792 which is less than .7916666 by .0000874; now the sine of 52° 21' exceeds that of 52° 20′ by 1777, therefore our angle 52° 20′ is too small by of a minute; that is, by 29" 77: whence, in strict exactness, angle B=52° 20' 29" and angle 4=37° 39′ 30′′ 177.

874

907 77779

870

The same by logarithms. Since log. sin B=log. AC+log. rad.-log. AB, to log. AC=log. 95..

Add log. radius = log. 10000000000=10.0000000

1.9777236

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9.8985424

Remains log. sin B=52° 20′=

Whence angle A=(90°—B=) 37° 40′ as before.

Instrumentally, by the Gunter.

Extend the compasses from 120 to 95 on the line (of numbers) marked Num. that extent will reach from (radius) 90° on the line (of sines) marked sin. to 52°=52° 20′ the angle B. We cannot find the side BC by this method, without anticipating case 4.

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2. In the right angled triangle ABC, given the hypothenuse AB=135, and the perpendicular AC=108, required the base

• An observation similar to that in the preceding note occurs here: the log. sine in the table which is the nearest to the above, is that of 52° 20′, viz. 9.8984944, but this is less than the above, being too small by 480, wherefore 52° 11' is too little for the angle B; now the difference between the log. sine of 52° 20′, and that of 52° 21' is 975, whence the above value of B is of a minute, or 29" too small; that is, the angle B=52° 20′ 29′′, and A=37° 39′ 30′′ by this mode of calculation.

It is worth while to observe, that the difference of about 3 of a second between this result, and that in the foregoing note, arises from the circumstance of the logarithms, as well as the sines, being approximations, and not absolutely exact.

When the sine, tangent, &c. found by operation is not in the table, 1. take the nearest from the table, and find the difference between that and the one found by operation; call this difference the numerator. 2. Find the difference of the next greater and next less than that found by operation, and call this difference the denominator. 3. Multiply the numerator by 60 and divide the product by the denominator, the quotient will be seconds, which must be added to, or subtracted from the degrees and minutes corresponding to the nearest tabular number, according as that number is less or greater than the number found by operation.

This rule will serve both for natural and logarithmic sines, tangents, &c. and likewise for the logarithms of numbers, observing in the latter case (instead of multiplying by 60) to subjoin a cipher to the numerator, and having divided by the denominator, the first quotient figure must occupy one place to the right of the right hand figure in the nearest tabular number, and be added, or subtracted, according as that number is too little, or too great.

BC, and the angles A and B? Ans. BC=81, ang. A=36° 52′, ang. B=53° 8'.

3. Given AB=291, BC=216, required the remaining side and angles? Ans. AC=195, ang. A=42° 5′, ang. B=47° 55′.

77. Case 2. Given the two sides AC and CB, to find the hypothenuse AB and the angles A and B.

First, (47.1.) AB=√AC+CB; whence AB is found.
Secondly, (Art. 65.) AC: CB:: radius: tangent ang. 4; or
CBX radius
; and by logarithms, log. tan. A=log. BC

tan A

AC

+log. rad.-log. AC, the angle A is found, both by natural numbers and logarithms, and the angle B=90°- A is likewise found.

EXAMPLES.-1. Given the side AC=123, and the side CB= 132, to find the hypothenuse AB and the angles A and B.

By calculation ↳.

First, AB=√AC)2+CB}2= √123)2+132)2= √/32553=

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1.0731707=natural tangent of 47° 1' ang. A, ang. B

(90°-4) 90°-47° 1' 42° 59′.

log. AB to log. CB 132=

Thirdly, by logarithms, log. tan. A=log. CB+log, rad.

Add log. radius 10000000000=10.0000000

And from the sum =

2.1205739

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Remains log. tan. ang. A=47° 1′=10.0306688

And ang. B=90°— A=42° 59′ as before.

Instrumentally.

Extend the compasses from 123 to 132 on the line (of numbers) marked Num. this extent will reach from (radius =) 45° on the line (of tangents) marked Tan. to 47° 1' the angle A.

In this and the following cases of right angled triangles, the construction is purposely omitted, it being perfectly easy and obvious, from what has been given on the subject in the Practical Geometry, near the end of part 8.

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