cot A, sec A, cosec A (or csc A).* Thus tan A is read "tangent A," and means "the tangent of the angle 4." The giving of names in (1) may be regarded as defining the trigonometric ratios. Definitions (1) may be expressed as follows:

These definitions can be given a slightly different form which is more general, and, accordingly, more useful in applications. In any right-angled triangle AMP (Fig. 2), M being the right angle, with reference to the angle A let MP be denoted as the opposite side, and AM as the adjacent side. Then these definitions take the form:

*The term sine first appeared in the twelfth century in a Latin translation of an Arabian work on astronomy, and was first used in a published work by a German mathematician, Regiomontanus (1436-1476). The terms secant and tangent were introduced by a Dane, Thomas Finck (1561-1646), in a work published in 1583. The term cosecant seems to have been first used by Rheticus, a German mathematician and astronomer (1514-1576), in one of his works which was published in 1596. The names cosine and cotangent were first employed by Edmund Gunter (1581-1626), professor of astronomy at Gresham College, London, who made the first table of logarithms of sines and tangents, published in 1620, and introduced the Gunter's chain now used in land surveying. The abbreviations sin, tan, sec, were first used in 1626 by a Flemish mathematician, Albert Girard (1590–1634), and those of cos, cot, appear to have been earliest used by an Englishman, William Oughtred (1574-1660), in his Trigonometry, published in 1657. These contractions, however, were not generally adopted until after their reintroduction by Leonhard Euler (1707–1783), born in Switzerland of Dutch descent, in a work published in 1748. They were simultaneously introduced in England by Thomas Simpson (1710-1761), professor at Woolwich, in his Trigonometry, published in 1748. [See Ball, A Short History of Mathematics, pp. 215, 367.] When first used these names referred, not to certain ratios connected with an angle, but to certain lines connected with circular arcs subtended by the angle. This is explained in Art. 79, which the student can easily read at this time. See Art. 80, Notes 2, 3.

[The word perpendicular is sometimes used instead of opposite side, and base instead of adjacent side.]

It is necessary that these definitions be thoroughly memorized.

EXAMPLES.

N.B. The student is requested to preserve the work and results of these Exs. for purposes of future reference.

1. In AMP (Fig. 2) give the trigonometric ratios of angle AMP. Note what ratios of angles A and P are equal.

2. In Figs. 45 a, 45 b, Art. 46, give the trigonometric ratios of the various acute angles.

3. Find the trigonometric ratios of the acute angles in the triangles in Exs. 8-10, Art. 8; Exs. 11-13, Art. 9; Exs. 2-6, Art. 11.

4. In a triangle PQR right-angled at Q, the hypotenuse PR is 10 in. long, and the side QR is 7. Find the trigonometric ratios of the angles P

and Q. Note what ratios of P and Q are equal.

5. For each of the angles in Ex. 4, and for each of any three of the angles in Ex. 3, calculate the following, and make a note of the result. [Let x denote the angle whose ratios are being considered.]

6. Make the same calculations for angle A in Fig. 2, Art. 12.

13. Definite and invariable connection between (acute) angles and trigonometric ratios. It is important that the following principles be clearly understood:

(1) To each value of an angle there corresponds but one value of each trigonometric ratio.

(2) Two unequal acute angles have different trigonometric ratios. (3) To each value of a trigonometric ratio there corresponds but one value of an acute angle.

(1) In Fig. 2, Art. 12, from any point S in AR draw ST perpendicular to AL. Let angle B (Fig. 3) be equal to A, and from any point G in one of the lines containing angle B draw GK perpendicular to the other line. Then, by definition (3), Art. 12,

But the triangles AMP, AST, BKG, are mutually equiangular. Hence the sides about the equal angles are proportional, and

Therefore all angles equal to A have the same sine. In like manner, these angles can be shown to have the same tangent, secant, etc.*

L1

(2) Let RAL and RAL, be any two unequal acute angles, placed, for convenience, so as to have a common vertex A and a common boundary line AR. From any point P on AL draw PM perpendicular to AR. Take AP1 = AP, and draw P1M1 perpendicular to AR. Then

* In Euclid's text on geometry, the properties of similar triangles are considered in Bk. VI. Pupils who study Euclid and have not reached Bk. VI. can be helped to understand these properties by means of a few exercises like those referred to in Ex. 3, Art. 12.

In a similar manner the other ratios can be shown to be respectively unequal.

Ex. In this construction AP1 is taken equal to AP. Why does this not affect the generality of the proof?

(3) This property follows as a corollary from (1) and (2).

The trigonometric ratios for angles from 0° to 90° are arranged in tables. In some tables the calculations are given to four places of decimals, in others to five, six, or seven places. There are also tables of the logarithms of the ratios (or of the logarithms increased by 10), which vary in the number of places of decimals to which the calculations are carried out. The student is advised to examine a table of the trigonometric ratios at this time. A good exercise will consist in finding the logarithms of some of the sines, tangents, etc., adding 10 to each logarithm, and comparing the result with that given in the table of Logarithmic sines, tangents, etc. [What are denoted as Natural sines and cosines in the tables, are merely the actual sines and cosines, which have been discussed above; the so-called Logarithmic sines and cosines are the logarithms of the Natural sines and cosines with 10 added.] A book of logarithms and trigonometric ratios is the principal help and tool in solving most of the problems in practical trigonometry; and hence, proficiency in using the tables is absolutely necessary. The larger part of the numerical answers in this book have been obtained with the aid of a five-place table. Those who use six-place or seven-place tables will reach more accurate results.

EXAMPLES.

1. Compare each of the ratios of RAL1 with the corresponding ratio of RAL.

2. Suppose that the line AR (Fig. 4) revolves about A in a counter-clockwise direction, starting from the position AM: show that, as the angle MAL

*These are usually called Logarithmic sines, tangents, etc.

increases, its sine, tangent, and secant increase, and its cosine, cotangent, and cosecant decrease. Test this conclusion by an inspection of a table of Natural ratios.

3. Find by tables, sin 17° 40', sin 43° 25' 10", sin 76° 43', sin 83° 20′ 25′′,

cos 72° 40' 30", tan 37° 40' 20",

cos 18° 10', cos 37° 40' 20", cos 61° 37', tan 79° 37' 30", cot 42° 30', cot 72° 25' 30". Log sin 37° 20', Log sin 70° 21' 30", Log cos 30° 20' 20", Log cos 71° 25', Log tan 79° 30' 20", Log cot 48° 20' 40".

4. Find the angles corresponding to the following Natural and Logarithmic ratios:

14. Practical problems. The problems in this article are intended to help the learner to realize more clearly and strongly the meaning and the usefulness of the ratios which have been defined in Art. 12. The student is earnestly recommended to try to solve the first three problems below without help from the book. He will find this to be an advantage, whether he can solve the problems or not. If he can solve them, then he will have the pleasurable feeling that he is to some extent independent of the book; and he will thus be encouraged and strengthened for future work. Should he fail to solve them, he will have the advantage of a closer acquaintance with the difficulties in the problems, and so will observe more keenly how these difficulties are avoided or Overcome. Throughout this course the student will find it to be of immense advantage if he will think and study over the subjectmatter indicated in the headings of the articles and make some kind of an attack on the problems before appealing to the book for help. If he follows this plan, his progress, in the long run, will be easier and more rapid, and his mental power more greatly improved than if he is content merely to follow after, or be led by, the teacher or author.