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80. Geometrical representation of the trigonometric functions. Let a circle of radius equal to unity be drawn. This circle is I called a unit-circle. Let the construction described in Art. 79 be made for each of the angles AOP1, AOP, AOP, .... In any circle the lines MP, MP2, MP3, are proportional to, and hence represent the sines of these angles,

the lines AT1, AT2, AT3, ***,

T3

B

T2

T1

A

MMM

represent the tangents of these angles,

the lines OT, OT1⁄2, OTз, ***,

represent the secants of these angles,

FIG. 70.

and so on for the other ratios. In the unit-circle, however, the measures of these lines, the radius being the unit of length, are the very same numbers as the respective ratios mentioned. In the unitcircle also, the linear measure of the arc is the same as the radian measure of the angle which it subtends. [See Art. 73, Note 2.]

SUGGESTED EXERCISES. (1) By means of the lines on the unit-circle, trace the changes in the trigonometric functions as the angle changes from 0° to 90°. Compare the results with those of Art. 77.

(2) For particular values of the angle AOP1, measure the lengths of the related lines on the unit-circle, and compare the results with the values given in the tables of natural sines and tangents.

NOTE 1. The origin of the terms circular functions, tangent, secant, is apparent from Art. 79.

NOTE 2. The name sine comes from the Latin word sinus, which was the translation of the Arabic word for this trigonometric function. The Arabic word for the sine resembled a word meaning an indentation or gulf.

NOTE 3. In trigonometry the Greeks used the whole chord P1Q instead of the half-chord or sine. For example, Ptolemy, the celebrated astronomer who flourished about 125-151 A.D., gave a table of chords in Book I. of the Almagest, his work on astronomy. The Hindoos, on the other hand, always used the half-chord or sine. The Arabian astronomer, Al Battani (or Albatagnius) (877-929), in his work The Science of the Stars, like the Hindoos determined angles "by the semi-chord of twice the angle," i.e. by the sine of the angle, taking the radius as unity. The translation of this work into Latin in the twelfth century introduced the word sine into trigonometry. The Hindoo sine was finally adopted in Europe in preference to the Greek chord in the fifteenth century. [See Art. 12, foot-note.]

81. Graphical representation of functions.

Graphical representation. The different values which a varying quantity takes, are often represented by means of a curve. Many illustrations can be given of the graphical representation of various things whose values can be denoted by means of numbers. For example, the curve in Fig. 71 shows the record of the barometer at Ithaca from May 22 to May 29, 1899.

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12 3 6 9 123 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 9 12 3 6 912
M. P.M. M. P.M. M. P.M. M. P.M. M. P.M. M. P.M.
Wednesday
May 24

Monday
May 22

Tuesday
May 23

Thursday
May 25
FIG. 71.

Friday
May 26

Saturday

May 27

M.
Sunday
May 28

P.M. M.

Monday

May 29

In this figure an hour is represented by a certain length, and the lengths representing hours are measured along a horizontal line. Each inch of height of the barometer is also represented by a certain length. At the points corresponding to the successive times perpendiculars are drawn, the lengths of which represent the heights of the barometer at the respective times. (In the figure the position of the horizontal line marked 29, represents the upper ends of heights of 29 inches.) The smooth curve drawn through the extremities of the perpendiculars is the barometric curve or curve of barometric heights for the period May 22 to May 29, 1899. This curve will give to most persons a clearer and more vivid idea of the range and variation of the height of the barometer during this period than a column of numbers of inches of heights is likely to give. If the scales used in representing the hours and the heights of the barometer were changed, then the curve would be somewhat altered, but its general appearance would remain the same.

The graph of a function. The graph of a function of x, say f(x), is obtained in the following way: Take a horizontal line X1OX; choose a point O, from which, distances representing the different values of x are measured along the line; measure positive values of x toward the right from O, and negative values toward the left. At particular points of X10X, at convenient distances apart, draw perpendiculars to represent the values of f(x) at the respective points. Draw the perpendiculars upward from XOX when the values of f(x) are positive, and downward when these values are negative. The smooth curve drawn through the extremities of these perpendiculars is the graph of f(x). The nearer the perpendiculars

are to one another, the better is the graph. For example, the function 2 x is represented (for certain values of x) by Fig. 72, the function2, by Fig. 73, the function √x, by Fig. 74. The pupil is advised to construct these graphs by following the method just described above.

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NOTE. The notion of representing a function by a curve is the fundamental notion in algebraic geometry, or, as it is usually termed, analytic geometry. This geometry was invented, in the form in which it is now known, by the philosopher and mathematician, René Descartes (1596–1650), and first published by him in 1637. This article may be regarded as a short lesson in the subject.

82. Graphs of the trigonometric functions.

take dis

Graph of sin 0. In order to draw the graph of sin tances, measured from O along the line XOX, to represent the number of radians in the angle 0. At points (not too far apart) on XOX draw perpendiculars to represent the sines of the angles corresponding to these points. The smooth curve drawn through the extremities of these perpendiculars will be the graph of the sine. Thus, for example, let a radian be represented by a unit length, and let the ratio unity be also represented by a unit length. Then (see Fig. 75) angle π (i.e. 180°) is represented by OM1 = 34. The perpendiculars at O and M1 are zero, since sin 0 = 0 and sin π = = 0. Erect perpendiculars equal to ..., sin 30°, sin 45°, sin 60°, sin 90°, ..., for instance, at the points corresponding to ......, ..., (i.e., 30°, 45°, 60°, 90°, ...,) respectively.

π

π ПП

6' 4' 3' 2'

Do the

same at points between L and M1, and draw the smooth curve OG,M, through the extremities of the perpendiculars. The successive perpendiculars from 7 to 2π are the same in length as those from 0 to π, but negative. From 2 to 4 the values of the sine are repeated in the same order as from 0 to 2 π. Hence, the graph of the sine can be obtained by merely successively reproducing the double undulation OG1M,G,M2, as indicated in Fig. 75. This is called the curve of sines, sine curve, or sinusoid.

2

NOTE 1. The unit circle (Art. 80) will be of service in drawing the graphs of the sine and the other trigonometric functions. For, if the scales for radians and ratios be those adopted above, then the horizontal distances from O will be equal to the lengths of the arcs (Art. 73, Note 2), and the lengths of the perpendiculars will be the lengths of the lines in the line definitions (Art. 79).

NOTE 2. If radians (i.e. 180°) be represented by a length different from that adopted in Fig. 75, then the graph of the sine will differ somewhat from Fig. 75, but its main features will be the same as in that figure. Figures 76 and 77 show portions of the graph of sin ✪ when π is represented on two other scales, while the sin (i.e. 1) is represented by a unit length. Hence the

π

2

curve of sines, or the sinusoid, may be defined as the curve in which horizontal distances measured on a certain line are proportional to an angle, and the perpendiculars to this line are proportional to its sine.

Ex. Draw the graphs for cos 0, tan 0, cot 0, sec 0, cosec 0.

Graph of cos 0. On using the same scales for radians and ratios as those adopted in Fig. 75, the graph of cos takes the form shown in Fig. 78. It is the same as the graph of sin 0 in Fig. 75 would be if O and the other points in XOX were all moved a distance toward the right. This might have been expected, since the sine of an angle is equal to the cosine of its complement. The values of the sine and the cosine alike range from +1 to 1.

π

Graph of tan 0. On using the same scales for radians and ratios as have been adopted in Fig. 75, the graph of tan takes the form shown in Fig. 79.

Graph of sec 0. On using the same scales for radians and ratios as have been adopted in Fig. 75, the graph of sec takes the form shown in Fig. 80.

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