5th, If A and B represent any two arcs, we have seen above, that tan A. cot A = rad2; and that tan B. cot B = rad2; therefore tan A cot A tan B. cot B. Whence cot A cot B:: tan B: tan A. And in a similar way we deduce cos A: sec B:: cos B: sec A; and sin A cosec B:: sin B: cosec A. 6th, As AD. DE = D B2, we have vers. suvers = sin2. 7th, The sine, tangent, &c. of an arc, which is the measure of any given angle, as ABC, is to the sine, tangent, &c. of any other arc, by which the same angle A B C may be measured, as the radius of the first arc to the radius of the second. E Let A C and M N each measure the angle B; CD being the sine, DA the versed sine, A E the tangent, and BE the secant of the arc AC; NO, O M, M P and BP the sine, versed sine, tangent, and secant of the arc M N. Then as ON, MP, DC, and A E, are parallel, we have CD: NO:: rad BC: rad B N; AE: MP::, or BE:BP: : rad B A and BC BD::BN: BO; or BA ВА : ВА BD BM: BM : MO; or BA:BM:: AD: MO. : : rad B M ; B OM D A Hence BD :: BM: BO. BO; or BA:AD::BM: Sth, It is often convenient in trigonometrical investigations, to use sines, tangents, &c. to the arcs of a circle whose radius is unity, as the resulting expressions are of a less complicated form. expressions may easily be adapted to any other radius. last figure, if BC = R, BN = 1, and DC = sine to whence BC (R): CD (sin) :: BN (1): NO sponding sine to radius unity. 'sin R و But such For, in the radius R; the corre Hence, when any formula has been investigated on the supposition that radius is unity, the formula may be adapted to any other radius sin tan sec R'R' R R, by substituting given expression. &c. for sine, tan, sec, &c. in the If the numerical values of the sines, tangents, &c. of every arc, were computed to a given radius, these numbers would exhibit the ratios of the sines, tangents, &c. to any other radius. A table containing such numbers is called a table of natural sines, tangents, &c.; and a table exhibiting the logarithms of those numbers is called a table of logarithmic sines, tangents, &c. Such logarithmic tables are generally computed to the large radius, 10000000000, that the logarithm of the smallest sine, tangent, &c. likely to be required in computation may not have a negative index. The logarithm of the radius in such tables is evidently 10, and the logarithm of rad2 is 20, and the logarithmic sine and cosine of any arc whatever is less than 10; but the logarithmic tangents, cotangents, secants, and cosecants, admit of all possible values from 0 to infinity, whatever be the numerical value of the radius. ON THE SIGNS OF TRIGONOMETRICAL QUANTITIES. WHEN geometrical quantities that are measured from some given point or line, are expressed analytically, they are considered as positive or negative; that is, as + or -, according as they lie on the same or opposite sides of that point or line, Thus, in the last figure but one, the sines are estimated from the diameter A E, and in the semicircle A IE they are considered as +; but as in the other semicircle E KA they fall on the other side of the diameter AE, they are then considered as The cosines, being estimated from the centre C, are considered as in the first quadrant AI; but as in the second quadrant IE, and the third quadrant E K, they fall on the other side of the centre C, they are then considered as; and again, in the fourth quadrant KA, they become +, as in the first quadrant. rad. sin As tan = COS the tangent is positive in the first quadrant, because sin and cos are then both positive. But in the second qua rad sin there is +. And in the fourth quadrant we have tan = fore in the fourth quadrant the tangent is The mutations of the signs of the cotangents, secants, and cosecants, may be traced in the same manner. As the versed sines are estimated from A, the extremity of the diameter, and all in one direction, they are positive in every quadrant of the circle. The following table exhibits the variations of the signs in each quadrant of the circle. Sin. Cos. Tan. Cot. Sec. Cosec. Vers. PROPOSITION I. ; The chord of 60° and the tangent of 45° are each equal to the radius the sine of 30°, the versed sine of 60°, and the cosine of 60° are each equal to half the radius; and the secant of 60° is equal to the diameter, or to double the radius. G Let D be the centre of a circle, A B an arc of 60°, and A C an arc of 45°. Draw the chord BA; bisect the angle A B D with the line B E, and draw the tangent A G, meeting the secants D B G and D C F in G and F. D B E Now (Theo. 63. Geo.) AB the chord of 60° has been shewn to be equal to the radius; and hence (Cor. 1. Theo.3. Geo.) BE bisects AD and is at right angles to it. Hence E A the versed sine of 60°, and DE the cosine of 60°, or the sine of 30°, are each equal to half the radius. And as B D is double of D E, by similar triangles D G the secant of 60° is double of D A the radius. Lastly, as the angle D A F is a right angle, and the angle A D F is 45°, or. half a right angle, the angle A F D is also half a right angle. Hence the angles A D F and A F D being equal, A F the tangent of 45° is equal to A D the radius. PROPOSITION II. If A be the greater, and B the less of two arcs, it is proposed to investigate the relation between their sines and cosines, and the sine and cosine of their sum, and of their difference. Let G be the centre of the circle, D C the greater are A, and D E or D F the less one B, join FE and G D then G D will bisect FE in I, and cut it also at right angles. Draw F M, IN, DO, and EP perpendicular to G C, and I H, EL perpendicular to F M. A + B; G M = cos A + D H K E NO PC Then DO = sin A, O G = cos A; FI or I E sin B; G I = cos B; F M = sin B ; EP = sin G A B; and G P = cos A B. Also as the angles H and K are right angles, and, from the parallel lines, HI and L E, the angles FIH and IE K are equal; and FI is also equal to IE, therefore the triangles FIH and IE K are identical, having F H equal to I K and H I equal to K E. Hence F M = IN+ F H and E P = IN - FH; GM = GN - IH and GP=GN+IH. Again, as the angles H IN and FIG are right angles, if the common angle HIG be taken from each, the remaining angles F I H and GIN will be equal. And as the angles H and N are right angles, the 1 triangle F I H is similar to the triangle G IN, and therefore similar also with the triangle G D O. And if radius be considered as unity, these formulas become From the above expressions, many curious and important formula may easily be deduced; the following are a few of the most useful. If the first expression be added to the second, we have sin A+B+ sin A B = 2 sin A. cos B If the second be subtracted from the first, we have B. These latter expressions being collected, we have sin A + B + sin A - B = 2 sin A: cos B We may observe on Formula 2, that as A is half the sum of A + B and A B; and B is half the difference of A + B and A - B; if therefore A + B be represented by P, and AB by Q, A will be To preserve uniformity of notation, we may in the above expressions put A and B instead of P and Q, and the expressions will then stand thus, If the first of Formula 3 be divided by the second, recollecting sin 2 2 A - B 2 ; or in words, as the sum of the sines of any two arcs is to the difference of the same sines, so is the tangent of half the sum of the arcs to the tangent of half their difference. |