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5th, If A and B represent any two arcs, we have seen above, that tan A .cot A = rad? ; and thåt tan B . cot B = rad”; therefore tan A . cot A = tan B . cot B. Whence cotA : 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 B’, we have vers suvers = sin?
7th, The sine, tangent, &c. of an arc, which is the measure of any given angle, as A B C, 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.
Let A C and M N each measure the angle B; C D being the sine, D A the versed sine, A E the tangent, and B E the
E secant of the arc AC; NO, OM, MP and BP
P the sine, versed sine, tangent, and secant of the arc MN. Then as O N, MP, DC, and A E, are parallel, we have CD:NO:: rad BC: rad B N; AE: MP::, or BE:BP:: rad B A : rad BM; B
OM DA and BC:BD::BN:BO; or BA : BD :: BM :BO. Hence BA: BA BD :: BM:BM - BO; or B A : AD:: BM : MO; or BA :BM :: AD: MO.
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. But such expressions may easily be adapted to any other radius. For, in the last figure, if B C = R, BN = 1, and DC = sine to radius R;
sin whence B C (R) : CD (sin) :: BN (1): NO
R sponding sine to radius unity.
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 R, by substituting
R' R' R
&c. for sine, tan, sec, &c. in the given expression.
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 radis 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.
Wuen 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 AI E they are considered as ti but as in the other semicircle E KA they fall on the other side of the diameter A E, 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 I E, 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 =
the tangent is positive in the first quadrant, because sin and cos are then both positive. But in the second qua
rad , sin drant cos is --, and sin t; therefore tan =
is In the
rad sin third quadrant sin and cos are both — ; therefore tan =
rad sin is +. And in the fourth quadrant we have tan =
therefore 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. 1st quadrant +
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 radįus.
Let D be the centre of a circle, A B an arc of 60°, and A C an arc of 45°, Draw the chord B A; bisect the angle
G A B D with the line B E, and draw the tangent AG, meeting the secants D B G and DCF in G and F.
F Now (Theo. 63. Geo.) A B 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
D E 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 AF 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.
If A be the greater, and B the less of two ares, 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
i then G D will bisect FE in I, and cut it also at right angles. Draw F M, I N, DO, and
E I E = sin B
cos B; F M = sin A + B; GM = cos A + B; EP = sin
NO PC A B; and GP= cos A B. Also as the angles H and K are right angles, and, from the parallel lines, H I and L E, the angles FIH and I E K are equal ; and FỊ is also equal to IE, therefore the triangles F IH and I E K are identical, having F H equal to IK and H I equal to K E. Hence FM = IN + FH and E P= IN - FH;
GM=GN – IH and GP=G N + I H. Again, as the angles H I N and F I G are right angles, if the common angle HIG he taken from each, the remaining angles FI H and GIN will be equal. And as the angles H and N are right angles, the triangle FI H is similar to the triangle GIN, and therefore similar also with the triangle GDO. We have therefore
DO.GI sin A,cos B GD:DO::GI:IN, or I N=
GO.FI cos A. sin B GD:GO::FI:FH, or FH =
GO.GI ços A.cos B GD:GO::GI: GN, or GN =
DO.FI sin A, sin B GD: DO::FI:IH, or I H =
sin A.cos B + cos A. sin B FM, or sin A + B =IN + FH =
sin A. cos B cos A, sin B EP, or sin A - B = IN- FH =
cos A.cos B sin A, sin B G M, or cos A + B =GN-HI =
rad GP, or cos A
cos A. cos B + sin A , sin B B = GN + H I =
rad And if radius be considered as unity, these formulas become
sin A + B = sin A . cos B + cos A . sin B
SCHOLIUM. 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. If the second be subtracted from the first, we have
sin A + B sin A - B = 2 cos A. sin B If the third and fourth be added together, we have
cos A + B + cos A B = 2 cos A. cos B If the third be subtracted from the fourth, we have cos A - B
cos A + B = 2 sin A , sin B These latter expressions being collected, we have
B cos A + B = 2 sin A . sin B
(2) 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 if therefore A + B be represented by P, and A B by Q, A will be P + Q
P Q represented by
and B by
and Formula 2 become 2
P+Q P Q sin P + sin Q = 2 sin
• COS 2
P + Q P Q sin P - sin Q = 2 cos
P+Q P cos P + cos Q = 2 cos
P+Q P-Q cos Q- cos P = 2 sin
2 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,
A + B A B sin A + sin B = 2 sin
A + B A B sin A · sin B = 2 cos
A + B A B cos A + cos B= 2 cos
2 A + B
A - B
(3) If the first of Formula 3 be divided by the second, recollecting sin
A + B
A + B Whence sin A + sin B : sin A sin B :: tan
2 A - B
; or in words, as the sum of the sines of any two arcs is to the 2 difference of the same sines, so is the tangent of half the sum of the arcs to the tangent of half their difference,