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PLANE TRIGONOMETRY.

DEFINITIONS.

I PLANE TRIGONOMETRY treats of the relations and calculations of the sides and angles of plane triangles.

2. The circumference of every circle (as before observed in Geom. def. 56) is supposed to be divided into 360 equal parts, called Degrees; also each degree into 60 Minutes, each minute into 60 seconds, and so on.

Hence a semicircle contains 180 degrees, and a quadrant 90 degrees.

3. The Measure of any angle (def. 57, Geom.) is an arc of any circle contained between the two lines which form that angle, the angular point being the centre; and it is estimated by the number of degrees contained in that arc.

Hence, a right angle being measured by a quadrant, or quarter of the circle, is an angle of 90 degrees; and the sum of the three angles of every triangle, or two right angles, is equal to 180 degrees. Therefore, in a right-angled triangle, taking one of the acute angles from 90 degrees, leaves the other acute angle; and the sum of two angles, in any triangle, taken from 180 degrees, leaves the third angle; or one angle being taken from 180 degrees, leaves the sum of the other two angles.

4. Degrees are marked at the top of the figure with a small, minutes with ', seconds with ", and so on. Thus, 57° 30′ 12′′, denote 57 degrees 30 minutes and 12 seconds.

5. The Complement of an arc, is what it wants of a quadrant or 90°. Thus, if AD be a quadrant, then BD is the complement of the arc AB; and, reciprocally, AB is the complement of BD. So that, if AB be an arc of 50°, then its complement BD will be 40°.

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6. The Supplement of an arc, is what it wants of a semicircle, or 180° Thus, if ADE be a semicircle, then BDE is the supplement of the arc AB; and, reciprocally, AB is the supplement of the arc BDE. So that, if AB be an arc of 50°, then its supplement BDE will be 130°.

7. The Sine, or Right Sine, of an arc, is the line drawn from one extremity of the arc, perpendicular to the diameter passing through the other extremity. Thus, BF is the sine of the arc AB, or of the arc BDE.

Corol. Hence the sine (BF) is half the chord (BG) of the double arc (BAG). 8. The Versed Sine of an arc, is the part of the diameter intercepted between the arc and its sine. So, AF is the versed sine of the arc AB, and EF the versed sine of the arc EDB.

9. The Tangent of an arc, is a line touching the circle in one extremity of that arc, continued from thence to meet a line drawn from the centre through the other extremity: which last line is called the Secant of the same arc. Thus, AH is the tangent, and CH the secant, of the arc AB. Also, -EI is the tangent, and CI the secant, of the supplemental arc BDE. And this latter tan

gent and secant are equal to the former, but are accounted negative, as being drawn in an opposite or contrary direction to the former.

10. The Cosine, Cotangent, and Cosecant, of an arc, are the sine, tangent, and secant of the complement of that arc, the Co being only a contraction of the word complement. Thus, the arcs AB, BD being the complements of each other, the sine, tangent or secant of the one of these, is the cosine, cotangent or cosecant of the other. So, BF, the sine of AB, is the cosine of BD; and BK, the sine of BD, is the cosine of AB: in like manner, AH, the tangent of AB, is the cotangent of BD; and DL, the tangent of DB, is the cotangent of AB: also, CH, the secant of AB, is the cosecant of BD; and CL, the secant of BD, is the cosecant of AB.

Corol. Hence several remarkable properties easily follow from these definitions; as,

1st, That an arc and its supplement have the same sine, tangent, and secant; but the two latter, the tangent and secant, are accounted negative when the arc is greater than a quadrant or 90 degrees.

2d, When the arc is 0, or nothing, the sine and tangent are nothing, but the secant is then the radius CA.-But when the arc is a quadrant AD, then the sine is the greatest it can be, being the radius CD of the circle; and both the tangent and secant are infinite.

3d, of any arc AB, the versed sine AF, and cosine BK, or CF, together make up the radius CA of the circle.-The radius CA, tangent AH, and secant CH, form a right-angled triangle CAH. So also do the radius, sine, and cosine, form another right-angled triangle CBF or CBK. As also the radius, cotangent, and cosecant, another right-angled triangle CDL. And all these rightangled triangles are similar to each other.

11. The sine, tangent, or secant of an angle, is the sine, tangent, or secant of the arc by which the angle is measured, or of the degrees, &c. in the same arc or angle.

12. The method of constructing the scales of chords, sines, tangents, and secants, usually engraven on instruments, for practice, is exhibited in the annexed figure.

13. A Trigonometrical Canon, is a table exhibiting the length of the sine, tangent, and secant, to every degree and minute of the quadrant, with respect to the radius, which is expressed by unity, or 1, and conceived to be divided into 10000000 or more decimal parts. And farther, the logarithms of these sines, tangents, and secants are also ranged in the tables; which are most commonly used, as they perform the calculations by only addition and subtraction, instead of the multiplication and division by the natural sines, &c. according to the nature of logarithms.

Upon this table depends the numeral solution of the several cases in trigono

Sines.

Secants.

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Chords.

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Tangents.

metry. It will therefore be proper to begin with the mode of constructing it, which may be done in the following manner:

PROBLEM I.

To find the sine and cosine of a given arc.

This problem is resolved after various ways. One of these is as follows, viz. by means of the ratio between the diameter and circumference of a circle, together with the known series for the sine and cosine, hereafter demonstrated. Thus, the semicircumference of the circle, whose radius is 1, being 3.141592653589793, &c., the proportion will therefore be,

as the number of degrees or minutes in the semicircle,

is to the degrees or minutes in the proposed arc,

so is 3.14159265, &c. to the length of the said arc.

This length of the arc being denoted by the letter a; also its sine and cosine by s and c; then will these two be expressed by the two following series, viz.

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EXAMPLE I.-If it be required to find the sine and cosine of one minute. Then, the number of minutes in 180° being 10800, it will be first, as 10800: 1 :: 3.14159265, &c.: 000290888208665 = the length of an arc of one minute. Therefore, in this case,

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EXAMPLE II. For the sine and cosine of 5 degrees.

Here, as 180°: 5o :: 3·14159265, &c. : •08726646 = a the length of 5 degrees. Hence, a = 08726646

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these collected, give s=08715574 the sine of 5o.

And, for the cosine, 1 = 1

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After the same manner, the sine and cosine of any other arc may be computed. But the greater the arc is, the slower the series will converge, in which case a greater number of terms must be taken to bring out the conclusion to the same degree of exactness.

Or, having found the sine, the cosine will be found from it, by the property of the right-angled triangle CBF, viz. the cosine CF = √ CB2 — BF2, or c = √T — s2.

There are also other methods of constructing the canon of sines and cosines, which, for brevity's sake, are here omitted.

PROBLEM II.

To compute the tangents and secants.

The sines and cosines being known, or found by the foregoing problem; the tangents and secants will be easily found, from the principle of similar triangles, in the following manner :—

In the first figure, where, of the arc AB, BF is the sine, CF or BK the cosine, AH the tangent, CH the secant, DL the cotangent, and CL the cosecant, the radius being CA, or CB, or CD; the three similar triangles CFB, CAH, CDL, give the following proportions:

1st, CF: FB:: CA: AH; whence the tangent is known, being a fourth proportional to the cosine, sine, and radius.

2d, CF: CB:: CA: CH; whence the secant is known, being a third proportional to the cosine and radius.

3d, BF: FC:: CD: DL; whence the cotangent is known, being a fourth proportional to the sine, cosine, and radius.

4th, BF: BC:: CD: CL; whence the cosecant is known, being a third proportional to the sine and radius.

Having given an idea of the calculation of sines, tangents, and secants, we may now proceed to resolve the several cases of Trigonometry; previous to which, however, it may be proper to add a few preparatory notes and observations, as below.

Note 1.-There are usually three methods of resolving triangles, or the cases of trigonometry; namely, Geometrical Construction, Arithmetical Computation, and Instrumental Operation.

In the First Method. The triangle is constructed by making the parts of the given magnitudes, namely, the sides from a scale of equal parts, and the angles from a scale of chords, or by some other instrument. Then, measuring the unknown parts, by the same scales or instruments, the solution will be obtained near the truth.

In the Second Method. Having stated the terms of the proportion according to the proper rule or theorem, resolve it like any other proportion, in which a fourth term is to be found from three given terms, by multiplying the second and third together, and dividing the product by the first, in working with the natural numbers; or, in working with the logarithms, add the logs. of the second and third terms together, and from the sum take the log. of the first term; then the natural number answering to the remainder is the fourth term sought. In the Third Method.-Or Instrumentally, as suppose by the log. lines on one side of the common two foot scales; Extend the compasses from the first term, to the second or third, which happens to be of the same kind with it; then that extent will reach from the other term to the fourth term, as required, taking both extents towards the same end of the scale.

Note 2.-In every triangle, or case in trigonometry, there must be given three parts, to find the other three. And, of the three parts that are given, one of

them at least must be a side; because the same angles are common to an infinite number of triangles.

Note 3.—All the cases in trigonometry may be comprised in three varieties only; viz.

1st, When a side and its opposite angle are given.

2d, When two sides and the contained angle are given.

3d, When the three sides are given.

For there cannot possibly be more than these three varieties of cases; for each of which it will therefore be proper to give a separate theorem, as follows:

THEOREM I.

When a side and its opposite angle are two of the given parts.

Then the sides of the triangle have the same proportion to each other, as the sines of their opposite angles have.

That is, As any one side,

Is to the sine of its opposite angle;

So is any other side,

To the sine of its opposite angle.

Demonstr.-For, let ABC be the proposed triangle, having AB the greatest side, and BC the least. Take ADBC, considering it as a radius; and let fall the perpendiculars DE, CF, which will evidently be the sines of the angles A and B, to the radius AD or BC. But the triangles ADE, ACF, are equiangular, and therefore AC: CF :: AD or BC : DE; that is, AC is to the sine of its opposite angle B, as BC to the sine of its opposite angle A.

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Note 1.-In practice, to find an angle, begin the proportion with a side opposite a given angle. And to find a side, begin with an angle opposite a given side.

Note 2.-An angle found by this rule is ambiguous, or uncertain whether it be acute or obtuse, unless it be a right angle, or unless its magnitude be such as to prevent the ambiguity; because the sine answers to two angles, which are supplements to each other; and accordingly the geometrical construction forms two triangles with the same parts that are given, as in the example below; and when there is no restriction or limitation included in the question, either of them may be taken. The degrees in the table, answering to the sine, is the acute angle; but if the angle be obtuse, subtract those degrees from 180o, and the remainder will be the obtuse angle. When a given angle is obtuse, or a right one, there can be no ambiguity; for then neither of the other angles can be obtuse, and the geometrical construction will form only one triangle.

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Required the other parts.

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