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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 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.
4. Degrees are marked at the top of the figure with a smallo, 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
D) 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 10
So that, if Al be an arc of 50°, then its complement BD :vill be 40°.
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 að 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 which
passes through the other extremity. Thus, BF is the sine of the arc AB, or of the supplemental arc BDE. 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, Er is the tangent, and ci the secant, of the supplemental arc
And this latter tangent 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,
Ist, 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 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 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
7° structing the scales of chords, sines, tangents, and secants, usually engraven on instruments, for practice, is exlibited in the annexed figure.
A Trigonoinetrical 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