PROBLEM XIX.

To determine a triangle; having given the base, the perpendicular, and the difference of the two other sides.

PROBLEM XX.

To determine a triangle; having given the base, the perpendicular, and the rectangle or product of the two sides.

PROBLEM XXI.

To determine a triangle; having given the lengths of three lines drawn from the three angles, to the middle of the oppo. site sides.

PROBLEM XXII.

In a triangle, having given all the three sides; to find the radius of the inscribed circle.

PROBLEM XXIII.

To determine a right-angled triangle; having given the side of the inscribed square, and the radius of the inscribed circle.

PROBLEM XXIV.

To determine a triangle, and the radius of the inscribed circle; having given the lengths of three lines drawn from the three angles, to the centre of that circle.

PROBLEM XXV.

To determine a right-angled triangle; having given the hypothenuse, and the radius of the inscribed circle.

PROBLEM XXVI.

To determine a triangle; having given the base, the line bisecting the vertical angle, and the diameter of the circum. scribing circle.

37

PLANE TRIGONOMETRY.

DEFINITIONS.

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 se. micircle 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.

DLH

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 49°.

6. The Supplement of an arc, is what it wants of a semicircle, or 180o. Thus, if ADE be a semicircle, then

E

I

K

F

G

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 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 cur the secant, of the arc ab. Also, EI is the tangent, and cr the secant, of the supplemental arc BDE. 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 important 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 co 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

adius ca, the tangent AH, and the secant CH, form a rightangled triangle CAн. 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 mea. sured, 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.

20

20

13. A Trigonometrical Canon, is a table showing the length of the sine, tangent, and secant, to every 30 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 of these sines, tangents, and secants, are also ranged in the

tables; and these 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. Such tables of log. sines and tangents, as well as the logs of common numbers, greatly facilitate trigonometrical computations, and are now very common. Among the most correct are those published by the author of this Course.

PROBLEM I.

To compute the Natural 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; and its sine and cosine by s and c; then will these two be ex. pressed by the two following series, viz.

EXAM. 1. If it be required to find the sine and cosine of 1 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,

EXAM. 2. For the sine and cosine of 5 degrees.

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