Trigonometry

“ 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. ”

“ 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. ”

“ 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. ”

“ Without some knowledge of trigonometry, you will not be able fully to understand this process; but, as it is in its nature somewhat curious, and is applied to various other geographical measurements, as well as to the determination of arcs of the meridian, I am desirous that you should understand its general principles. Let us reflect, then, that it must be a matter of the greatest difficulty, to execute with exactness the measurement of a line of any great length in one continued direction on the earth's surface. ”

“ The unit of circular measure is too large to be convenient for practical purposes, but its use introduces a simplification into the series in analytical trigonometry, owing to the fact that the sine of an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, are ultimately in a ratio of equality. ”

“ The whole doctrine of astronomy rests upon trigonometry, a branch of the science of mathematics which teaches us, having two sides and one angle, or two angles and one side, of a triangle given us, to construct the whole. ”

“ Since any plane triangle can be divided into right-angled triangles, the solution of all plane triangles can be reduced to that of right-angled triangles; moreover, according to the theory of similar triangles, the ratios between pairs of sides of a right angled triangle depend only upon the magnitude of the acute angles of the triangle, and may therefore be regarded as functions of either of these angles. The primary object of trigonometry, therefore, requires a classification and numerical tabulation of these functions of an angular magnitude ”

“ 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. ”

“ Further, the reduction of the doctrine of the sphere to a quantitative form needed for astronomical purposes, required the formation of a spherical trigonometry, which {56} was also achieved by Hipparchus. Thus both plane and spherical trigonometry, which are parts of the highly abstract and simple science of extension, remained undeveloped until the less abstract and more complex science of the celestial motions had need of them. ”

“ Therefore, if we could easily measure ∠O and could measure the distance x, we could find the height of a [Pg 236] building y. In trigonometry we use a transit for measuring angles, but it is easy to measure them with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held against it, so as to find the angle of elevation of a building. ”

“ 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 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. ”

“ To obviate the inevitable inaccuracies attending this graphic method and as a corroboration of the ship’s position, the changes of latitude and longitude involved in each alteration of course are daily calculated by plane trigonometry, such calculations being materially abbreviated by the use of the Traverse Table, which is a tabulated expression of the solutions of right-angled plane triangles. ”

“ Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical trigonometry. Also, everything relating to change of systems of axes, as for instance in the kinematics of a rigid system, where we have constantly to consider one set of rotations with regard to axes fixed in space, and another set with regard to axes fixed in the system, is a matter of troublesome complexity by the usual methods. ”

“ The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle is the angle subtended at the centre of a circle by an arc equal in length to the radius. ”

“ 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. ”

“ In trigonometry, is a right line raised perpendicularly on the extreme of the diameter, and continued to a point, where it is cut by a secant, that is, by a line drawn from the centre, through the extremity of the arch, whereof it is the tangent. ”

“ Trigonometrical curve, a curve one of whose coördinates is a trigonometric function of the other. — Trigonometrical function. See under Function. — Trigonometrical lines, lines which are employed in solving the different cases of plane and spherical trigonometry, as sines, tangents, secants, and the like. These lines, or the lengths of them, are trigonometrical functions of the arcs and angles to which they belong. ”

“ All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meridians and parallels within the space which is to be represented in a plane. Then the azimuth is represented unaltered, and any spherical distance z is represented by ρ. Thus we get all the points of intersection transferred to the representation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation. ”

“ It ought to be added that, wonderful as is the extent of physical phenomena open to our investigation, intellectual phenomena are yet vastly more extensive. Of this I might present one satisfactory proof were space available by pointing out that the mathematical functions employed in the calculations of physical science form an infinitely small fraction of the functions which might be invented. Common trigonometry consists of a great series of useful formulæ, all of which arise out of the relation of the sine and cosine expressed in one equation, sin 2x + cos 2x = 1. ”

“ He wrote a treatise in twelve Books on Chords in a Circle, equivalent to a table of trigonometrical sines. For calculating arcs in astronomy from other arcs given by means of tables he used propositions in spherical trigonometry. ”

“ TRIGONOMETRY 571 connected with the circle. We may easily show from the definitions that cos 2 (a; + iy) + sin* (x + iy) = 1, cosh 2 y - sinh - y 1 ; cos (ar + iy) = cos x cosh y - 1 sin x sinh y, sin (x + iy) = sin x cosh y + 1 cos x sinh y, cosh (a + |3) = cosh a cosh /3 + sinh a sinh /3, sinh (a + /3) = sinh a cosh - cosh a sinh /3. These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert. Expan- If we equate the coefficients of 71 on both sides of equation (13) , sionofan we set . . ”

“ The crystal is considered to be placed inside a sphere from the centre of which normals are drawn to all the faces of the crystal. The points at which these normals intersect the surface of the sphere are called the poles of the faces, and by these poles the positions of the faces are fixed. The poles of all faces in the same zone on the crystal will lie on a great circle of the sphere, which are therefore called zone-circles. The calculation of the angles between the normals of faces and between zone-circles is then performed by the ordinary methods of spherical trigonometry. ”

“ A. D. 100) extant in Arabic; Book I of this work contains propositions about spherical triangles corresponding to the main propositions of Euclid about plane triangles (e.g. congruence theorems and the proposition that in a spherical triangle the three angles are together greater than two right angles) , while Book III contains genuine spherical trigonometry, consisting of ‘Menelaus’s Theorem’ with reference to the sphere and deductions therefrom. ”

“ Similarly it can be shown 564 TRIGONOMETRY that nir + a includes all the angles which have the same tangent as a. Rela- From the Pythagorean theorem, the sum of the squares of the tions be- projections of any straight line upon two straight lines at right tween angles to one another is equal to the square on the projected line, trigono- we get sin 2 et + cosset = 1, and from this by the help qf the definitions metrical of the other functions we deduce the relations l + tan a a=sec 2 o, func- l + cot 2 a = cosec 2 a. ”

“ 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 . ”

“ The Indians did not apply their trigonometrical knowledge to the solution of triangles; for astronomical purposes they solved right-angled plane and spherical triangles by geometry.
The Arabs were acquainted with Ptolemy’s Almagest, and they probably learned from the Indians the use of the sine. ”

“ The Indians, who were much more apt calculators than the Greeks, availed themselves of the Greek geometry which came from Alexandria, and made it the basis of trigonometrical calculations. The principal improvement which they introduced consists in the formation of tables of half-chords or sines instead of chords. ”

“ GIBSON, WILLIAM (1720–1791) , self-taught mathematician, born at Boulton, near Appleby, Westmoreland, in 1720, worked on a farm from childhood, and afterwards obtained a farm of his own at Hollins, near Cartmel Fell, Lancashire. He received no education whatever in youth, but in early manhood taught himself to read a book on arithmetic, and developed an extraordinary power of working out sums of all kinds in his head. He afterwards taught himself writing, and studied geometry, trigonometry, algebra, and astronomy, in all of which he proved himself an expert. ”

“ We will consider the tangent as a secant joining two points infinitely near to each other; and then, designating by dy and dx the infinitely small differences of the co-ordinates of those two points, the elementary principles of geometry will immediately give the equation t = dy/dx for the trigonometrical tangent of the angle which is made with the axis of the abscissas by the desired tangent, this being the most simple way of fixing its position in a system of rectilinear co-ordinates. ”

“ 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 ”

“ 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. ”