24. Numb. 32 log. 1.5051500 numb. 009 log. — 3·9542425 3 3d pow. 32768 log. 4.5154500 000000729 log. 3 -7.8627275 NOTE. After multiplying the negative index, the carriage to it from the logarithm must be subtracted from the product. If the positive index be used, 10 times the name of the power lessened by 1 must be taken from the index of the power. 25. Number 0437 log. — 2·6404814, or 8-6404814 TO EXTRACT THE ROOT OF A NUMBER BY LOGARITHMS. Divide its logarithm by the name of the root: the quotient is the logarithm of the root. NOTE. If the given number be a decimal, and its index positive, prefix the name of the root lessened by 1 to the index, before dividing. If the index be negative, add to it the least number that will make the sum divisible by the name of the root: the quotient is the index of the root; but in dividing the logarithm, the number added only is to be considered as the index. 14335 4. 5. sixth root of 2. ninth of '0375. Ans. 947848. 1.1224628. 12 ciphers before 1466. -627537. compound interest of £67.495 for 5 years, at 4 per cent. Ans. £15, 14s. rate of comp. int. at which £136.782 will, in 5 years, amount to £173.564. Ans. 4.64. time in which £53.5 will amount to £76-36, at 3 per cent. comp. int. Ans. 10.342 years. PLANE TRIGONOMETRY. TRIGONOMETRY is the method of determining the sides and angles of triangles, and of expressing them in known measures. This is done by means of the ratios which certain straight lines in and about the circle have to its radius. DEFINITIONS. A 1. The SINE BG of an arc AB, is a straight line drawn from B, one of its extremities, perpendicular to the diameter AE, which passes through the other. 2. The VERSED SINE AG of an arc AB, is the portion of the diameter AE upon which the sine is perpendicular, between the sine and the arc. 3. The TANGENT AF of an arc AB, touches the circle at A, one of the extremities of the arc, and meets at F the diameter MB, which passes through the other extremity B. 4. The SECANT CF of an arc AB, is a straight line drawn from C the centre, to F the farthest extremity of the tangent. 5. The sine, versed sine, tangent, and secant of an arc AB, are called the sine, versed sine, tangent, and secant of the angle ACB measured by that arc to the radius AC. 6. The SUPPLEMENT of an arc AB, or of an angle ACB, is the difference between it and 180°. Thus BE or AM is the supplement of AB, and BCE or ACM the supplement of ACB. Cor. 1. An arc, or angle, and its supplement, have the same sine, tangent, and secant; for BG is the sine of BE or BCE, AF the tangent of AM or ACM, and CF the secant of AM or ACM. Cor. 2. The versed sine EG of BCE, together with AG the versed sine of ACB, is equal to the diameter AE. 7. The COMPLEMENT of an arc AB, or angle ACB, is the difference between it and 90°. Thus BD or BCD is the complement of AB or ACB. 8. The sine, versed sine, tangent, and secant of the complement of an arc or angle, are called the cosine, coversed sine, cotangent, and cosecant of the arc or angle. Thus BH or CG is the cosine of AB or ACB, DH is its coversed sine, DK its cotangent, and CK its cosecant. Cor. 1. The cosine CG, together with the versed sine AG, is equal to the radius AC. Cor. 2. The sine BG of an arc AB, is half of BL, the chord of BAL the double of AB. Cor. 3. The radius is equal to the sine or versed sine of 90°, and to the tangent or cotangent of 45°. NOTE 1. In what follows, we generally use sin. for sine, cos. for cosine, tan. for tangent, sec. for secant, ver. for versed sine, cov. for coversed sine, cot. for cotangent, cosec. for cosecant, cho. for chord, R. or rad. for radius, and D. or dia. for diameter. NOTE 2. For the purpose of performing arithmetically the operations of trigonometry, a circle has been selected of which the radius is very large, such as 100000, &c.; and the sines, tangents, &c. have been calculated for every second of the quadrant of such a circle, and arranged in tables; and from these the sines, tangents, &c. for arcs of other circles may be found by proportion. OF THE TABLES OF SINES, TANGENTS, AND SECANTS. The common tables have the degrees at the top, and the minutes on the left side, when the degrees are less than 45°; but if greater, the degrees are marked at the bottom, and the minutes on the right side. The logarithms of the natural sines, tangents, &c. have been taken, and placed in similar tables. These form the tables of artificial sines, tangents, &c. which supply the place of the natural ones in the same way that the logarithms supply the place of natural numbers. 1. Required the artificial sine of 37° 23′ 12′′. Turn to the page which has 37° at the top, and come down the column titled Sine at the top, to the line that has 23' on the left side, and you will find 9.7832922, the sine of 37° 23′; and the difference between it and the sine of 37° 24′ is 1653. Then as 60" is to 12", so is 1653 to 331, the proportional difference for 12", which, added to 9.7832922, gives 9.7833253, the sine of 37° 23′ 12′′. 2. Required the degrees and parts of a degree of which 10-2738462 is the artificial tangent. Look for the nearest tangent 10-2737163, and because it is titled Tang. at the bottom, take the degrees at the foot, and the minutes on the right side, where are found 61° 58'. The difference between this tangent and the one above it is 3046, and the difference between it and the given one is 1299; herefore 3046: 1299:: 60′′: 26′′, so that 10-2738462 is the angent of 61° 58′ 26′′. 3. Natural sine of 57° 26' 20". 4. Artificial cosine of 67° 31′ 40′′. Ans. ·8428179. 9-5823310. 10-5456998. 41° 36′ 36". 70° 28′ 20′′. SOLUTION OF RIGHT-ANGLED TRIANGLES. THE first thing to be done in resolving right-angled triangles is to make one of the sides the radius of a circle, the centre of which is at an acute angle, and thus to determine what the other sides would be in that circle. If from the centre A, with the radius AC, the arc CD be described, then BC will be the sine of CAB, and AB its cosine. But if the centre be at C, and the circle pass through A, then AB is the sine of C, and BC its cosine. Hence when the hypotenuse is radius, the other sides are the sines of their opposite angles, or the cosines of their adjacent angles. Again, if from the centre A, with the radius AB, the are BE be described, then BC is the tangent of A, and AC is its secant. Suppose ACB any angle, and AB an are described with the radius of the circle, from which the sines, tangents, &c. in the tables were calculated; then BF is the sine in the tables, CF the cosine, AG the tangent, and CG the secant in the tables Let CEH be a right-angled triangle. If CE be radius, EH will be the side of C, and CH ite ousine, But the triangles CEH, CBF, being sir, CE: EM :: CB: BF; that is, as CE is to EH, so is the radius of the tables to the sine of C in the tables. In like mouse to CH as the radius to the cosine in the tabuer. In the mouse way it may be shown, that if CDK were the triuge, bud (1) the radius, CD is to DK as the rative to the tongue, sk, the tables, and that DC is to CK as the radiu i wyk of C in the tables; so that after determining the woude of the sides of the triangle, any two sites are whakak names in the tables The terms of the proportion, however, must be so arranged, that the thing required shall be the last term, thus: And these three are all the variations which are requisite. But the student should accustom himself to state them without hesitation. 1. In the triangle ABC, right-angled at B, are given the hypotenuse AC 324 feet, and the angle BAC 48° 17; to find the base AB, and perpendicular BC. ⚫ NOTE. When one of the acute angles is known, the other is got by subtracting that one from 90o. Å A B If AC be radius, and A the centre, CB is the sine of A, and AB its cosine. Wherefore, R: sin. A :: AC: CB, and R : cos. A :: CA : AB. Sin. A 48° 17′ log. 9-8729976 cos. A log. 9.8231138 log. 2.5105450 AC 324 Sum Radius CB 241.85 12.3835426 log. 10-0000000 2-5105450 12.3336588 10-0000000 log. 2.3835426 AB 215-6 log. 2:3336588 2. Given DE 1254 feet, and the angle D 51° 19'; to find the hypotenuse DF, and the perpendicular EF. DE being radius, EF is the tangent and DF the secant of D. R: tan. D:: DE: EF. Tan. D 51o 19'-R. log. 0-0965445 D F DE 1254 log. 3.0982975 3. Given the angle G 43° 38', and the opposite side HK 186 feet; to find the hypotenuse GK, and the base GH. K 4 This may be wrought as the last, by first finding GH GKH. Or, GK being radius, KH is the sin. G; and GH being radius, HK is tan. G. |