Page images






1. 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, and each minute into 60 Seconds, and so on. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees.

3. The Measure of an 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 the 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. VOL. II. B

4. Degrees

[ocr errors]

4. Decrees are marked at the top of the figure with a Str.?!" minutes with , seconds with", and so on. Thus, 3*.com , denote 57 degrees 30 minutes and 12 seconds.

The Connement oi an arc, is futai i wants oia ouairant or 90°.

KI ins. ii ar be a anatrant, then BD is

contenien; oi the arc Ab; and, If sna'N, Àt is the complement of

so that, ti Ak be an arc of 50%, 401 1*. (fe::ion: Rp will be 40.

To statement of an art, is T i ra, z, wants a semicircis, or 180°. 'stats i Antir a semicircle, then BDE is the supplement of Phr Ara Aki and, reciprocally, AB is the supplement of the ፡፡ ከ[. in that, if AR be an art of 50', then its suppleinni ent will be 1300.

7. Sine, or Right Sino, of an arc, is the line drawn fron anr extremit of the arc, perpendicular to the diameter m. base di pes throngl. the other extremity. Thus, BF is the une a tho ari. Ah, em of the supplemental arc BDE. Hence #hr in Hi! is daalf the chord (BG) of the double arc

[ocr errors][ocr errors]
[ocr errors]

$ The Porse. Sine aan ani, i the part of the diameter UITEIT, ad harwron the an ancians sine. So, AF is the versed und wurde un Ah, and IT Ihr versei sine of the arc EDB.

a. Thir Tangan a an ari, k a line touching the circle in one EUISTIT I Thar art, canranned from thence to meet a toe drawn from the cenere trynch the other extremity; which last time is called the Sacar of the same arc. Thus, In is the tangent, and on the secani, of the art AB. Also, El is the tangan, and a the socant, of the supplemental arc EDE.

And this latter tangent and setan 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 aros 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 Ar, is the cosine of BD; and BK, the sine of R1), is the cosine of as: in like manner, Ah, the tangent of B, is the cotangent of bd; and DL, the tangent of

the cotangent of AB : also, ch, the secant of AB,
osecant of BD; and cl, the secant of BD, is the co-
f AD.



. 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, the least it can be. As the arc increases from 0, the sines, tangents, and secants, all proceed increasing, till the arc becomes a whole quadrant AD, and 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, the tangent AH, and the secant ch, form a rightangled triangle cay. 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 col: And all these right-angled 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 con

7o structing 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 showing 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, with any number of ciphers. The logarithms

Vers. Sin of these sines, tangents, and secants, are also ranged in the B2






[ocr errors]


[ocr errors]
[ocr errors]




« PreviousContinue »