PLANE TRIGONOMETRY.

DEFINITIONS AND PRINCIPLES.

1. PLANE Trigonometry teaches how to determine, from proper data, the sides and angles of plane rectilineal triangles, by means of the analogies of certain right lines, described in, and about a circle.

2. Every triangle contains six parts, viz. three sides, and three angles; any three of these, whereof one (at least) is a side, being given, the remaining three may be found.

3. The sides of plane rectilineal triangles are estimated in feet, yards, fathoms, chains, &c. or by abstract numbers: and each of the angles, by the arc of a circle included between the two legs; the angular point being the centre.

4. It has already been observed (Art. 237. part 8.), that the whole circumference is supposed to be divided into 360 degrees, each degree into 60 minutes, each minute into 60 seconds, &c.; as many degrees, minutes, and seconds therefore, as are con tained in the arc intercepted between the legs of an angle, so many degrees, minutes, and seconds, that angle is said to measure; and, note, in the following definitions, whatever is affirmed of an arc, is likewise affirmed of the angle (at the centre,) which stands on that arc.

5. Draw any straight line AC; from C as a centre with the distance CA, describe the circle AEN; produce AC to L, and through the centre C draw ECK perpendicular to AL; in the arc EA take any point B, join BA, BE, and BC, and produce the latter to N; through A and B draw AT, BD each parallel to CE, and produce them to S and G; join CG, and produce it to R and S, produce CB to T, through E and B draw REH, MËB; each parallel to CA, and join BL, MN; then since TA, BD are both parallel to EC, they are parallel to one another (30. 1.), and both perpendicular to C4 (29. 1.); for a like reason EH and FB

a An easy tract on Plane Trigonometry may be found in Ludlam's Rudiments of Mathematics. Mr. Bridge's lectures on the same subject, published in 1810, is likewise a neat end useful work.

are parallel, and both perpendicular to EC, and BD=FC, and

Thus, the arc BE is the complement of the arc AB; and the angle BCE is the complement of the angle ACB '.

8. The difference of any arc from a semicircle, or 180°, or of any angle from two right angles, is called THE SUPPLEMENT of that arc or angle.

Thus, the arc BL is the supplement of the arc AB, and the angle BCL of the angle ACB c.

9. THE CHORD of an arc is a straight line drawn from one end of the arc to the other.

b In like manner AB is the complement of BE, and the angle ACB of the angle BCE. The name complement likewise applies to the excess of an arc above a quadrant, or of an angle above a right angle; thus EB is the complement of the arc BML, and of the angle BCL; but in most practical questions it is usually restrained to what an arc or acute angle wants of 90o.

The arc AB is likewise the supplement of the arc BML, and the angle ACB of the angle BCL. The term supplement means also the excess of an are above a semicircle, thus the arc AB is the supplement of the arc AMN. The difference of an arc from the whole circumference is termed its supplement to a circle,

Thus, the straight line AB is the chord of the arc AB, or of the angle ACB.

Cor. The chord of 60° is equal to the radius (cor. 15.4.); and the chord of 180° is the diameter.

10. THE CO-CHORD of an arc, is the chord of the complement of that arc.

Thus, the straight line BE (or the chord of the arc BE) is the co-chord of the arc AB, or of the angle ACB.

11. THE SUPPLEMENTAL CHORD of an arc, is the chord of its supplement.

Thus, BL (or the chord of the are BML) is the supplemental chord of the arc AB, or of the angle ACB.

Cor. Hence it appears that the chord of any arc, is likewise the chord of its supplement to a whole circle; also that the chord can never exceed the diameter (15. 3.)

Thus, BL is not only the chord of the arc BML, but also of the arc BKL.

12. THE SINE of an arc, is a straight line drawn from one end of the arc, perpendicular to the diameter which passes through the other end of the arc.

Thus, BD is the sine of the arc AB, and of the angle ACB. Cor. Hence the sine of an arc, is the same as the sine of its supplement, for BD is not only the sine of the arc AB, but also of the arc BML; for it is drawn from one extremity B, (of the arc BML,) perpendicular to the diameter AL, passing through the other extremity L.

13. THE CO-SINE of an arc, is that part of the diameter (passing through the beginning of the arc,) which is intercepted between the sine and the centre, and is equal to the sine of the complement of that arc.

Thus, CD is the co-sine of the arc AB, and of the angle ACB; and it is equal to BF (34.1) the sine of BE, which is the complement of AB.

Cor. Hence the sine of a quadrant, or of a right angle (is not only equal to, but) is the radius; and the co-sine of a quadrant or right angle is nothing.

Thus, if the point B be supposed to move to E, the arc AB will become AE, the sine of which is EC; and the point D coinciding with C, the co-sine CD will vanish.

Hence also the sine or co-sine can never exceed the radius. 14. THE VERSED SINE of an arc, is that part of the diameter which is intercepted between the beginning of the arc and its şine.

Thus, DA is the versed sine of the arc AB, and of the angle ACB; and AP is the versed sine of the arc ABM, and of the angle ACM.

Cor. Hence the versed sine of an arc less than a quadrant, is equal to the difference; and of an arc greater than a quadrant, to the sum of the co-sine and radius.

Thus, AD (the versed sine of AB) —C4—CD, and AP (the versed sine of ABM) =CA+CP.

Hence also the versed sine (being always within the circle,) can never exceed the diameter, (15.3.)

15. THE CO-VERSED

SINE of an arc, is the

versed sine of its complement.

Thus, EF is the coversed sine of the arc AB, and of the angle ACB.

Cor. Hence the co

versed sine is equal to the excess of the radius, above L the sine.

16. THE TANGENT of an arc, is a straight line at right angles to the diameter, passing through one end of the arc, and meeting a diameter produced through the other end of the arc.

Thus, AT is the targent of the arc AB, and of the angle ACB.

Cor. Hence a tangent may be of any magnitude (according to the magnitude of its arc) from nothing to infinity. Hence also the tangent of 45° is equal to the radius (6. 1.)

17. THE CO-TANGENT of an arc, is the tangent of the complement of that arc.

Thus, EH (the tangent of EB) is the co-tangent of the arc AB, and of the angle ACB.

18. THE SECANT of an arc, is a straight line drawn from the centre, through the end of the arc, and produced till it meet the tangent.

Thus, CT is the secant of the arc AB, and of the angle ACB. Cor. Hence a secant can never be less than the radius, but it increases (as the arc increases) from the radius to infinity, 19. THE CO-SECANT of an arc is the secant of its complement.

Thus, CH (the secant of EB,) is the co-secant of the arc AB, and of the angle ACB a,

THE VARIATIONS, AND ALGEBRAIC SIGNS, OF THE TRIGONOMETRICAL LINES IN THE FOUR QUAD

RANTS.

20. If the sine, co-sine, tangent, co-tangent, secant, co-secant, versed sine, and co-versed sine for every arc in the first quadrant AE be drawn, they will serve for the three remaining quadrants EL, LK, KA, that is, for the whole circle, as will be shewn further on; but previous to this, it will be necessary to suppose the point B to coincide with A, and to move from thence round the whole circumference, and this will lead us to explain the manner of applying the algebraic signs and to the lines peculiar to Trigonometry.

21. When the point B coincides with A, the arc AB will ➡o̟, and the points D and T will coincide with A; wherefore AT=0, BD=0, DA=0, CB and CD each radius; that is, the tangent, sine, and versed sine, (of o degrees, or) at the beginning of the quadrant will be nothing, and the secant and cosine will be radius.

* Some of the trigonometrical lines received their names from the parts of an archer's bow, to which they bear a similitude; thus, ARC comes from arcus, a bow; CHORD from chorda, the string of a bow; SAGITTA (now generally called the versed sine) from sagitta, an arrow; SINE from sinus, the bosom, alluding to that part of the chorda or string, which is held near the breast in the act of shooting, the sine being half the chord of double the arc. The prefix co is an abbreviation of the word complement; thus co-sine, co-tangent, &c. imply complement sine, complement tangent, &c. or the sine, tangent, &c. of the complement of a given arc.