PROBLEM XIV. To determine a triangle; having given the base, the perpendicular, and the ratio of the two sides. PROBLEM XV. To determine a right-angled triangle; having given the hypothenuse, and the side of the inscribed square. PROBLEM XVI. To determine the radii of three equal circles, described in a given circle, to touch each other and also the circumference of the given circle. PROBLEM XVII. In a right-angled triangle, having given the perimeter, or sum of all the sides, and the perpendicular let fall from the right angle on the hypothenuse; to determine the triangle, that is, its sides. PROBLEM XVIII. To determine a right-angled triangle; having given the hypothenuse, and the difference of two lines drawn from the two acute angles to the centre of the inscribed circle. PROBLEM XIX. To determine a right-angled triangle; having given the side of the inscribed square, and the radius of the inscribed circle. PROBLEM XX. To determine a right-angled triangle; having given the hypothenuse, and the radius of the inscribed circle. PROBLEM XXI. To divide a line of 10 inches in extreme and mean ratio. PROBLEM XXII. To add to it a segment, such that the rectangle under the whole line thus increased, and the part of it increased, shall be to the square of the difference of the two segments into which the line is now divided, as 12: 5. PROBLEM XXIII. FROM a given point in the plane of a given circle, to draw a chord of a given length a, the distance of the point from the centre of the circle being b. PROBLEM XXIV. GIVEN the adjacent sides a, b, and the diagonal of a parallelogram, to find the other diagonal. PROBLEM XXV. GIVEN the chords of two arcs of a given circle, to find the chord of their sum, and the chord of their difference. PROBLEM XXVI. To divide the base, a, of a triangle into two segments proportional to the sides, b, c. PROBLEM XXVII. Two circles being given which touch one another inwardly; to describe a third circle that shall touch both the former, and also the right line passing through their centres. PROBLEM XXVIII. HAVING given the lengths of two chords which intersect at right angles, and the distance of their point of intersection from the centre; to find the diameter of the circle. PROBLEM XXIX. GIVEN, to determine the area of the triangle, and the lengths of its sides, the three perpendiculars from the angles upon the opposite sides. PROBLEM XXX. GIVEN the four sides of a quadrilateral inscribed in a circle, to find its diagonals, and the radius of the circle in which it is inscribed. PROBLEM XXXI. Supposing the town A to be 30 miles from B, B 25 miles from C, and C 20 miles from A; if a house be erected to be equally distant from each of those towns, what will the distance be? Ans. 15.11856 miles. PROBLEM XXXII. The three chords of three arches completing a semicircle being given, 3, 4, and 5 respectively; required the diameter. Ans. 8.05581. PLANE TRIGONOMETRY. DEFINITIONS. 1. PLANE TRIGONOMETRY treats of the relations subsisting between the sides and angles of plane triangles, and the method of calculating from a sufficient number of data the remaining ones, whether sides or angles. 2. The circumference of every circle (as stated in Geom. Def. 59) 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. 60, Geom ) is an arc of any circle contained between the two lines which form that angle, the angular point being the centre of the circle; and it is estimated by the number of degrees and subordinate parts of a degree 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, one of the acute angles taken 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. 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 its difference from 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 40°. 6. The Supplement of an arc, is its difference from a semicirle 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. Κ So that, if AB be an arc of 50°, then its supplement BDE will be 130° †. H * It is necessary that the student should keep this distinctly in his mind :-that angles as they are considered in Geometry do not enter into calculation, but only the arcs of circles which are their measures. It is, however, very customary to write either arc or angle indiscriminately; though only the arc is really meant. + Considering defs. 5 and 6 generally, the excess of an arc above a quadrant or of a semicircle, are respectively called the complement and the supplement of that arc. In these cases the complement and supplement are to be considered as negative, or affected with the sign whilst in those defined in the text they are positive or +. 7. The 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). Or the sine may be thus defined: it is the altitude of the point B above the line CA; or the depression of G below the line CA. If the angle which is considered positive lie on the upper side of CA, the angle which lies below it will be negative; and the converse. The arcs which measure these will also be denominated in the same manner. The signs and, as in common algebra, are said to designate these positions of the arcs, and the arcs and their sines. 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, EI is the tangent, and CI 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 AC: also, CH, the secant of AB, is the cosecant of BD; and CL, the secant of BD, is the cosecant of AB. Thus, again, DK, the versed sine of BD, is the coversed sine of AB. Analogous to the definition of the sine that of the cosine will naturally become, the distance of the point B from the line CD (drawn at right angles to CA) on either side of it. It will be positive when to the right, and negative when to the left of that line, the commencement of the arc being as before at A, or of its angle at CA, and proceeding in the direction ABD. These are also designated by + and prefixed to the name or symbol of the line itself. 11. Most frequently an angle, or its measure the arc, is denoted by a single letter in the same manner as in algebra, as a, b, c, &c. or A, B, C, &c. Sometimes the Greek letters are also used for the same purpose. 12. The names of the lines given in the preceding definitions are written in an abbreviated form, thus-sin. for sine, cos. for cosine, tan. for tangent, cot. for cotangent, sec. for secant, cosec. for cosecant, versin. (or vers.) for versed sine, covers. for coversed sine, and rad. for radius. 13. The sines, cosines, tangents, &c. though formerly designated (and occasionally so still with advantage) by single letters, and those of most frequent occurrence by their initials, as s, c, t, &c.—are now expressed by their abbreviated names prefixed to the symbol of the arc to which they belong; as sin. a for the sine of the arc a, tan. B, for the tangent of the arc or angle B, sec. a for the secant of the arc x, &c. Corol. Hence several important consequences easily follow from these definitions, and the properties of the figure to which they belong; 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 of 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. 14. When the sides and angles of a triangle ABC are spoken of, the angles are usually called A, B, C, and the opposite sides denoted by a, b, c ; A being the angle opposite to the side a, (or BC of the figure), B opposite to b, and C opposite to c. This method greatly facilitates recollection, and has 15. The method of constructing the scales of chords, sines, tangents, and secants, usually engraven on instruments, for practice, is exhibited in the annexed figure. 16. 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, or else to 1, with any number of ciphers affixed to it. 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. Sines. 10 Secants many advantages. 20 40 60 Note.-When the angles of a triangle are under consideration, describe arcs of circles from each angle, as a centre, and intercepted by the sides, with the same radius (that radius being the unit of length by which it is proposed to measure the sides of the triangles): then the lines drawn in the resulting figure to fulfil the conditions of the several definitions, are those whose lengths are VOL. I. 80 Chords bo 20 RA B M T G I 75 Tangents. |