Sine (trigonometry)

“ Moreover, he used sometimes also the abbreviations which are current at the present time, namely sin=sine, tan=tangent, sec=secant. We know that the Circles of Proportion existed in manuscript many years before they were published. The symbol sv for sinus versus occurs in the Clavis of 1631. The great importance of well-chosen symbols needs no emphasis to readers of the present day. With reference to Oughtred’s trigonometric symbols. Augustus De Morgan said:
This is so very important a step, simple as it is, that Euler is justly held to have greatly advanced trigonometry by its introduction. ”

“ The new height will be the sine of the new angle , or, stating it as an equation,
;
and subtracting from this the first equation gives
.
The quantity on the right-hand side is the difference between two sines, and books on trigonometry tell us how to work this out. For they tell us that if and are two different angles,
.
If, then, we put for one angle, and for the other, we may write
But if we regard as indefinitely small, then in the limit we may neglect by comparison with , and may also take as being the same as . ”

“ If the measure of the sine line of the angle under consideration is equal to the measure of the sine line of one of the other angles they both must be acute, for it is impossible for them both to be obtuse, as we have shown; and it is likewise impossible for one to be acute and the other obtuse, for if this were so these angles would be like an acute and obtuse angle that share one sine line, the two of which already total 180 degrees without the inclusion of the third angle of the triangle, and all three angles of a triangle must total no more than 180 degrees (as was written in section 63) . ”

“ We see, then, that in our applications, we should distrust those formulas which give an angle by its sine or cosine. The tangent 39 being alone exempt from these difficulties, we should seek, as far as possible, to resolve all questions by means of it. Thus, let us suppose that we know the hypothenuse and one of the sides of a right-angled triangle, the direct determination of the included angle will be given by a cosine, which will be wanting in exactness if the hypothenuse of the triangle does not differ much from the given side. ”

“ The first tables of logarithmic sines and tangents were constructed by Edmund Gunter (1581-1626) , professor of astronomy at Gresham College, London; he was also the first to employ the expressions cosine, cotangent, and cosecant for the sine, tangent, and secant of the complement of an arc. A treatise by Albert Girard (1590-1634) , published at The Hague in 1626, contains the theorems which give areas of spherical triangles and polygons, and applications of the properties of the supplementary triangles to the reduction of the number of different cases in the solution of spherical triangles. ”

“ Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by Δ the area, of the triangle. We might, as in § 6, take CA, CB as axes of x and y, inclined at an angle C. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin C from CA and CB. Call these β and α respectively. ”

“ A closer competitor for the honor of first using these trigonometric abbreviations is Richard Norwood in his Trigonometrie, London, 1631, where s stands for sine, t for tangent, sc for sine complement (cosine) , tc for tangent complement (cotangent) , and sec for secant. Norwood was a teacher of mathematics in London and a well-known writer of books on navigation. Aside from the abbreviations just [38] cited Norwood did not use nearly as much symbolism in his mathematics as did Oughtred. ”

“ If a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, right-angled at A, then . This is frequently called "Geber's Theorem." Radical and bold as were his innovations in spherical trigonometry, in plane trigonometry he followed slavishly the old beaten path of the Greeks. ”

“ The trisector of an angle, if he demand attention from any mathematician, is bound to produce, from his construction, an expression for the sine or cosine of the third part of any angle, in terms of the sine or cosine of the angle itself, obtained by help of no higher than the square root. The mathematician knows that such a thing cannot be ”

“ The figure drawn on the blackboard is the sensible object. The conception of a triangle is the intelligible object formed by abstraction, and universal. In this conception are contained the general notions of a point, a line, an angle; and in these notions are involved several self-evident principles or axioms. From these are demonstrated the various mathematical propositions of trigonometry. It is easy to see that, in the intellectual process of the pupil’s mind, the genesis and development of the act of cognition of mathematical truth is precisely what has been above described. ”

“ We have only to select a number of such triangles and a system of trigonometry ensues, in which base, perpendicular, and hypotenuse of every triangle is a whole measure without fractions, and in which the nomenclature for every angle is clear and simple. ”

“ In finding the base we may pursue another method, namely:—
Add the logarithm of the versed sine of the given angle to the logarithms of the given sides, and the sum will be the logarithm of the difference between the versed sine of the difference of the sides and the versed sine of the base required. This difference being consequently known, add to it the versed sine of the difference of the sides, and the sum will be the versed sine of the base required. ”

“ It is useless to consider here separately spherical trigonometry, which cannot give rise to any special philosophical consideration; since, essential as it is by the importance and the multiplicity of its uses, it can be treated at the present day only as a simple application of rectilinear trigonometry, which furnishes directly its fundamental equations, by substituting for the spherical triangle the corresponding trihedral angle. ”

“ Furthermore, I have learned that while positive assertions on any given subject are often true, error creeps in when simple aspects are denied in order to emphasise the more complex, or vice versa. A trigonometrical sine, for instance, may be expressed in terms of imaginary exponentials in a way familiar to all mathematical students ”

“ Trigonometry, in its essential form of showing how to deduce the values of the angles and sides of a triangle when other angles and sides are given, is an invention of the Greeks. It found its origin in the computations demanded for the reduction of astronomical observations and in other problems connected with astronomical science; and since spherical triangles specially occur, it happened that spherical trigonometry was developed before the simpler plane trigonometry. ”

“ Every new class of surfaces which may be considered would have in this extended sense a trigonometry of its own, which would consist in an investigation of the nature and properties of the functions necessary for the measurement of the sides and angles of triangles bounded by geodesics drawn on such surfaces. ”

“ This method has its roots in the profoundest depths of the human intellect, and hence is of far higher importance than the most ingenious analytical processes which are applicable only to a particular case. True, trigonometry is analytical geometry; as taught in the gymnasia, trigonometry, like stereometry, as both these terms indicate, has to do rather with mensuration, and its use is restricted to a certain class of problems. ”

“ If seven-figure logarithms do not give sufficiently accurate results, it is usual to have recourse to ten-figure tables; with one exception, there exist no tables giving 8 or 9 figures. The exception is John Newton's Trigonometria Britannica (London, 1658) , which gives logarithms of numbers to 100,000 to 8 places, and also log sines and tangents for every centesimal minute (i.e., the nine-thousandth part of a right angle) , and also log sines and tangents for the first three degrees of the quadrant to 5 places, the interval being the one-thousandth part of a degree. ”

“ At a first thought it might appear as if these eight movements had nothing in common with trigonometry, but a very fundamental connection may be seen as follows: If the vertex of the angle A is the center of a square and the initial line of A coincides with a line of symmetry of the square, the operations of taking the complement and the supplement of A correspond to movements transforming the square into itself. ”

“ If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB, RACB, RAD, and the points A and B approach and meet: I say, that the ultimate form of these evanescent triangles is that of similitude, and their ultimate ratio that of equality. For while the point B approaches towards the point A, consider always AB, AD, AR, as produced to the remote points b, d, and r, and rbd as drawn parallel to RD, and let the arc Acb be always similar to the arc ACB. Then supposing the points A and B to coincide, the angle bAd will vanish ”

“ From double this, namely 4357140, subtract 693147, and there will remain 3663993. The arc corresponding to this is 1° 28’, and the number put for the sine is 25595; but this is also the versed sine of 13°.⁠⁂
5.⁠Given two arcs, to find a third whose sine shall be equal to the sum of the sines of the given arcs. ”

“ In toothed armatures, the broken surface of the core, and the still greater reaction from the armature current, may produce wide variations from the sine law, the general tendency being to give the E.M.F. curve a more peaked form. The great convenience of the assumption that the E.M.F. obeys the sine law has led to its being very commonly used as the basis for the mathematical analysis of alternator problems; but any deductions made from this premiss require to be applied with caution if they are likely to be modified by a different shape of the curve. ”

“ Then, with my compass at E, I sight another line at an angle of sixty degrees from my last. I am making what is called an equilateral triangle; that is, a triangle with equal sides and equal angles. Each angle must measure sixty degrees. With two angles and one side, we can always get the other two sides; and the other angle will be where those two sides meet. They will meet at C. Now, since the sides are of equal length, the distance from D to C is the same as from D to E,—that is, four rods and thirty links, just the distance we wish to go; C, then, is the place for your corner stake. ”

“ In Chapter XV. we found that there was such a function–namely, the sine (or the cosine also) which possessed this property. So, without further ado, we may infer that the solution will be of the form . However, let us go to work.
Multiply both sides of the original equation by and integrate, giving us , and, as ,
being a constant. Then, taking the square roots,
But it can be shown that (see p. 171)
whence, passing from angles to sines,
where is a constant angle that comes in by integration. ”

“ One should also know that when three angles are known the length of the sides cannot be deduced, for only their proportion to each other can be determined.
Section 110 In a right triangle there are only three necessary arrangements of quantities. Since the right angle is known only two other quantities need be known: another angle and side or two sides [or two angles which is not mentioned] . ”