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PLANE TRIGONOMETRY.

DEFINITIONS.

1. In every plane triangle there are six parts: three sides and three angles. These parts are so related to each other, that when one side and any two other parts are given, the remaining ones can be obtained, either by geometrical construction or by trigonometrical computation.

2. Plane Trigonometry explains the methods of computing the unknown parts of a plane triangle, when a suf ficient number of the six parts is given.

3. For the purpose of trigonometrical calculation, the circumference of the circle is supposed to be divided into 360 equal parts, called degrees; each degree is supposed to be divided into 60 equal parts, called minutes; and each minute into 60 equal parts, called séconds.

Degrees, minutes, and seconds, are designated respectively, by the characters '". For example, ten degrees, eighteen minutes, and fourteen seconds, would be written 10° 18' 14".

4. If two lines be drawn through the centre of the circle, at right angles to each other, they will divide the circumference into four equal parts, of 90° each. Every right angle then, as EOA, is measured by an arc of 90°; every acute angle, as BOA, by an arc less than 90°; and every obtuse angle, as FOA, by an arc greater than 90°.

5. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc EB is the complement of AB. The sum of an arc and its complement is equal to 90°.

6. The supplement of an arc is what remains after subtracting the arc from 180°. Thus, GF is the

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supplement of the arc AEF. supplement is equal to 180°.

The sum of an arc and its

7. The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, BD is the sine of the arc AB.

8. The cosine of an arc is the part of the diameter intercepted between the foot of the sine and the centre. 'Thus, OD is the cosine of the arc AB.

9. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. Thus, AC is the tangent of the arc AB.

10. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. Thus, OC is the secant of the arc AB.

11. The four lines, BD, OD, AC, OC, depend for their values on the arc AB and the radius OA; they are thus designated :

sin AB for BD

cos AB for OD

tan AB for AC

sec AB for 00

Let the lines ET and IB

12. If ABE be equal to a quadrant, or 90°, then EB will be the complement of AB. be drawn perpendicular to OE.

Then,

ET, the tangent of EB, is called the cotangent of AB;
IB, the sine of EB, is equal to the cosine of AB;
OT, the secant of EB, is called the cosecant of AB.

In general, if A is any arc or angle, we have,

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arc is equal to the cosine of its supplement.*

Furthermore, AQ is the tangent of the arc AF, and

OQ is its secant: GL is the

tangent, and OL the secant

But since AQ is equal to

of the supplemental arc GF. GL, and OQ to OL, it follows that, the tangent of an arc is equal to the tangent of its supplement; and the secant of om arc is equal to the secant of its supplement.*

TABLE OF NATURAL SINES.

14. Let us suppose, that in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent, have been calculated for every minute or second of the quadrant, an arranged in a table; such a table is called a table of ines and tangents. If the radius of the circle is 1, the table is called a table of natural sines. A table of natural sines, therefore, shows the values of the sines, cosines, tangents, and cotangents of all the arcs of a quadrant, which is divided to minutes or seconds.

If the sines, cosines, tangents, and secants are known for arcs less than 90°, those for arcs which are greater can be found from them. For if an arc is less than 90°, its supplement will be greater than 90°, and the numerical values of these lines are the same for an arc and its supplement. Thus, if we know the sine of 20°, we also know the sine of its supplement 160°; for the two are equal to each other. The Table of Natural Sines is not given, as it is much easier to make the computations by the Table which we are about to explain.

*These relations are between the numerical values of the trigonometrical lines; the algebraic signs, which they have in the different quadrants, are not considered.

TABLE OF LOGARITHMIC SINES.

15. In this table are arranged the logarithms of the numerical values of the sines, cosines, tangents, and cotangents of all the arcs of a quadrant, calculated to a ra dius of 10,000,000,000. The logarithm of this radius is 10 In the first and last horizontal lines of each page, are writ ten the degrees whose sines, cosines, &c., are expressed on the page. The vertical columns on the left and right, are

columns of minutes.

CASE I.

To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle.

16. If the angle is less than 45°, look for the degrees in the first horizontal line of the different pages: when the degrees are found, descend along the column of minutes, on the left of the page, till you reach the number showing the minutes: then pass along a horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be: the number so indicated is the logarithm sought. Thus, on page 37, for 19° 55', we find,

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17. If the angle is greater than 45°, search for the de grees along the bottom line of the different pages: when the number is found, ascend along the column of minutes on the right hand side of the page, till you reach the number express ing the minutes: then pass along a horizontal line into the column designated tang, cot, sine, or cosine, as the case may be: the number so pointed out is the logarithm required.

18. The column designated sine, at the top of the page, is designated by cosine at the bottom; the one designated tang, by cotang, and the one designated cotang, by tang.

The angle found by taking the degrees at the top of the page, and the minutes from the left hand vertical column, is the complement of the angle found by taking the degrees

at the bottom of the page, and the minutes from the right hand column on the same horizontal line with the first. Therefore, sine, at the top of the page, should correspond with cosine, at the bottom; cosine with sine, tang with cotang, and cotang with tang, as in the tables (Art. 12).

If the angle is greater than 90°, we have only to sub tract it from 180°, and take the sine, cosine, tangent, or cotangent of the remainder.

The column of the table next to the column of sines, and on the right of it, is designated by the letter D. This column is calculated in the following manner.

Opening the table at any page, as 42, the sine of 24° is found to be 9.609313; that of 24° 01', 9.609597: their difference is 284; this being divided by 60, the number of seconds in a minute, gives 4.73, which is entered in the column D.

Now, supposing the increase of the logarithmic sine to be proportional to the increase of the arc, and it is nearly so for 60", it follows, that 4.73 is the increase of the sine for 1". Similarly, if the arc were 24° 20′, the increase of the sine for 1", would be 4.65.

The same remarks are applicable in respect of the column D, after the column cosine, and of the column D, between the tangents and cotangents. The column D, between the columns tangents and cotangents, answers to both of these columns.

Now, if it were required to find the logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before; then, multiply the corresponding tabular difference by the seconds, and add the product to the number first found, for the sine of the given arc.

Thus, if we wish the sine of 40° 26' 28".

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