NOTE. Before proceeding further, the student should be able to draw with ease a right-angled triangle, having been given: (a) The hypotenuse and a side; (b) the two sides about the right angle; (c) the hypotenuse and one of the acute angles; (d) one of the sides about the right angle and the opposite angle; (e) one of the sides about the right angle and the adjacent angle. It is here taken for granted that these problems have been considered in a course in plane geometry or in a course of geometrical drawing.

EXAMPLES.

N. B. The student is advised to do Exs. 1-6 carefully, and to preserve the results, for they will soon be required for purposes of illustration.

1. Draw to scale the triangles considered in Exs. 8, 9, 10, Art. 8, and Exs. 11, 12, 13, Art. 9, and measure the angles.

2. Make drawings, on two different scales, of a right-angled triangle whose base is 20 ft. and adjacent acute angle is 55°. In each drawing measure the remaining parts and thence deduce the unknown parts of the original triangle. In each drawing calculate the ratios specified in Ex. 8, Art. 8.

3. Same as Ex. 2, for a right-angled triangle whose hypotenuse is 30 ft. and angle at base is 25°.

4. Same as Ex. 2, for a right-angled triangle whose base and perpendicular are 30 ft. and 45 ft. respectively.

5. Same as Ex. 2, for a right-angled triangle whose hypotenuse is 60 ft. and base is 45 ft.

6. Same as Ex. 2, for a right-angled triangle in which the base is 50 ft. and the angle opposite to the base is 40°.

7. What angles of a whole number of degrees can easily be constructed geometrically without the aid of the protractor? Make the constructions.

12. Trigonometric ratios defined for acute angles. The ratios referred to at the beginning of Art. 8 will now be explained so far as acute angles are concerned. (Before proceeding, the student

should glance over the work on Exs. 8-10, Art. 8; Exs. 11-13, Art. 9; Exs. 1–6, Art. 11.) Let A be any acute angle. In either one of the lines containing the angle take any point P and let fall a perpendicular PM to the other line. The three lines AP, AM, MP, can be taken by twos in three different ways, and hence six ratios can be formed with them, namely:

It is shown in Art. 13 that each of these ratios has the same value as in Fig. 2, no matter where the point P is taken on either one of the lines bounding an angle which is equal to A. For the sake of convenience of reference, each one of these six ratios is given a particular name with respect to the angle A. Thus :

These six ratios are known as the trigonometric ratios of the angle A. According to the definition of a ratio (Art. 8) they are merely numbers. For brevity they are written sin A, cos A, tan A,

* In Chapter V. the trigonometric ratios are defined for angles in general. The definitions give in this article will be found to follow immediately from those given in Art. 40.

cot A, sec A, cosec A (or csc A).* Thus tan A is read "tangent A," and means "the tangent of the angle A." 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 APM. 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 R. Note what ratios of P and R 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

MP ST KG

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.*

R

Li

(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 PM1 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.