To the arc FC, it follows that C D will be its sine ; F G its tangent; A G its secant; D A its cosine; E T its cotangent; and A Tits cosecant. Since F C and E C are complements of each other; also, since C B = DA, and CD= BA, therefore CB, the sine of E C = DA, the cosine of FC Thus: The sine of an angle is equal to the cosine of its complement ; the cosine to the sine of its complement; the tangent to the cotangent of its complement. 9. The lines to which names (as sines, tangents, etc.) have been given in the preceding sections, are sufficiently correct as geometrical lines, appertaining to the arc E C, or angle E A C, which corresponds to the radius A E; thus, generally, lines so defined refer to circular arcs and the angles they subtend (or measure). To a circle less, or greater, than that in the fig., there would be no change in the angle by the diminution, or extension, of radius A C and AE; but the sines, tangents, &c., would differ considerably in length, and in fact before the value of the lines could be found we must know the length of the radius proper to the lines. There is a way out of this difficulty. Trigonometry deals with the angles and sides of triangles, irrespective of arcs that measure angles; and since an invariable angle must have an invariable sine, cosine, tangent, etc., the difficulty is overcome by always considering radius as unity, or the abstract number 1, and treating the lines on the basis of ratios. To this method of treatment belong the trigonometrical functions, as distinguished from the geometrical definitions and their consequences. Ratio.-It is well that you should have a clear notion of the meaning of the term ratio, which is commonly used in Trigonometry. Instead of saying that A is to B in the proportion of 3 to 5, we say “in the ratio of 3 to 5," and it is usually expressed as a fraction. The ratio of one magnitude to another is independent of the kind of magnitudes compared : thus, one may contain the other, or the fifth, or twentieth, or hundredth part of the other the same number of times, whether they be lines, or surfaces, or solids, or again, weights or parts of duration. It is required only for the comparison that the magnitudes be of the same kind, containing the same magnitude, each of them a certain number of times, or a certain number of times nearly. Upon these numbers, and upon these only, the ratio depends. In brief then, ratio may be defined as the relative values of two quantities of the same kind, or the number of times that one contains the other. TRIGONOMETRICAL TABLES.—There are two kinds of Trigonometrical tables that may be used in the computation of the sides and angles of a triangle. One kind contains the sines, cosines, tangents, etc., of every degree and minute of the quadrant, calculated as decimal fractions, to radius unity, or the abstract number 1. The sines, tangents, &c., of these tables are called NATURAL SINES,—and thus we speak of natural sines, natural cosines, natural tangents, &c., but they are not generally used in naviga tion, and some works do not even contain them. In the use of these tables, the product of two quantities is got only by multiplication, and the quotient by division, which is sometimes tedious. Tables constructed to radius i are called natural to distinguish them from another description of trigonometrical tables which are called logarithmic. The logarithmic sines, cosines, tangents, &c., are constructed on the base of the natural sines, cosines, tangents, &c. As in the latter case, all the sines and cosines, all the tangents from o° to 45°, and all the cotangents from 45° to 90°, are less than unity (or I), the logarithms of these quantities have negative characteristics or indices. But, in order to avoid the necessity of entering negative numbers, the logarithmic tables are constructed by adding 10 to every index, and so registered. By this contrivance, addition does the work of multiplication, and division that of subtraction, and thus, in trigonometrical calculations, by the aid of logarithms of numbers on the same construction, much time and labour are saved. The tables here referred to are Log. Sines, etc., and Logarithms; when using them it is better to distinguish the logarithms of numbers as log. ; those of sines, cosines, and tangents, etc., as L sin., L cos., L tan., etc. TRIGONOMETRICAL FUNCTIONS OF AN ANGLE In Trigonometry we deal with sides and angles of a triangle, and the trigonometrical functions of an angle are independent of the magnitude of the radius. Take several right-angled triangles (see Fig. I), as B AC, D A E, FAG, HA K, etc., which have the same acute angle at A common to each triangle ; it is certain that the triangles are not equal, since they differ in magnitude, one from another ; but they are nevertheless in all respects similar, since, the acute angle A, appertaining equally to all the triangles, is due to the ratio between the sides being constant; the magnitude of the sides is not in question; the condition lies in this, that whatever part A C is of A E, the same part must C B be of E D, etc., and so forth; Fig. 1. or, on the basis of ratios E B F H CB: CA CB ED: EA = GF: GA GF: GA = KH: KA KH KA Or, since the ratio is generally expressed in the form of a fraction. To impress upon you more clearly still the idea of what is meant by the connection between the angles and the ratios of a triangle ; in the above fig. right-angled at B, say we have _ C and _ A each equal to 45°; this would not arise from the opposite sides being each 10 feet, or each 50 feet, or each 100 feet, but from the equality of the lengths. So also if ZC were 30°, as would happen if the hypotenuse were 100 feet, and the side opposite ZC only 50 feet, then <C would be 30°, not owing to the magnitude of the sides, but because the hypotenuse was double the side opposite ZC. Hence The angles of a triangle depend not upon the absolute length of the sides of the triangle, but upon their RELATIVE lengths, that is, upon the ratios existing between them. On page 41 the sines, cosines, tangents, &c., are taken to be lines connected with an arc, but in practice they are considered as quantities corre sponding to certain ratios, called trigonometrical functions of an angle. In the annexed figure (2) the three sides of the right-angled triangle, by taking the sides two and two, give six ratios as follow А B of which the last three are the inverse of the first three; and each of these ratios measures and determines the angle A, inasmuch as one variable Fig. 2. quantity which measures another will also determine it, if a determinate value of one necessarily corresponds to a determinate value of the other; and conversely." The definitions that the ratios, taken in the above order, give to the angle A, are in succession, since the right-angle B 90°; and A + C 90°; consequently C = (90° — A) being the complement of A; and the sine of an angle equals the cosine of its complement, etc., etc. These remarks, set out in tabular order, are more clearly seen below You should study these carefully, and be able to write them off rapidly, for on these six ratios depend the whole of the formulæ of Trigonometry. It not unfrequently happens that, in cases where the angle A is taken as contained by the base and hypotenuse, and subtended by the perpendicular, the trigonometrical ratios are given as Now the side that subtends the right angle is invariably called the hypotenuse, but it is the more frequent and better method to consider the remaining sides as sides, without any characteristic distinction beyond the literal one. From the trigonometrical ratios are readily deduced. RULES RELATIONS BETWEEN THE SIDES AND ANGLES.—As the six ratios determine the angle, they also give six equations through which the value of the respective sides may be found. In collating these (see Fig. 2, page 44), the angles are indicated by the capital letters A, B, C, and the corresponding sides opposite to them by the italic letters a, b, c; then we have Ratios, as Equations, or C as the Consequences. complement of A. 6 (1) C (5) (4) Cot. A tan. C; c = a cot. A = a tan. C b с 6 6 = a cosec. A = a sec. C (90° — A). Also by Euclid, Book I., prop. 47– a (7) Since b = a2 + c2, therefore b = Va? + a? = (b + a)(b -- a) But the methods of computation (7 and 8) give no saving of figures. The foregoing include all cases of finding either angle or any side in a right-angled triangle and are adapted to logarithmic computation. NOTE.-When two quantities are put together without any sign between them, they are multiplied together, thus in (1) b sin. A means b multiplied by sin. A. When quantities are within brackets without any sign between the brackets, the quantity within the one bracket is to be multiplied by the quantity within the other bracket, thus in (8) (6+c) (6-c) means that b+c is to be multiplied by 6-c. The sides and angles of a triangle make up the six parts of the triangle; and if any three of these six parts, excepting the three angles, be given, the remaining three may be found by calculation. In a right-angled triangle the right angle is always known, hence it is sufficient for the determination that any two of the other five parts (excepting the two acute angles) be given. It is obvious from the simplest principles of Geometry that the three angles alone cannot determine the other three parts, viz., the three sides, since all equiangular triangles are alike, in respect to the equality of the angles, but the sides may differ ; hence an infinite variety of triangles may be constructed with the same three angles. In a right-angled triangle, therefore, the given parts may be either GENERAL RULE TO FIND A SIDE.—In (1) to solve the triangle, we have to find the other two sides and the remaining acute angle. Let us first find one of the sides. Write down the side required, and underneath it put the given side so as to form a fraction; we shall then on reference to the figure see what trigonometrical ratio of the given angle has been made. Now put the sign of equality between the fraction and the trigonometrical ratio; the resulting equation will give the formula by which the required side is found. To exemplify this, suppose in the Fig. 2, p. 44, the side b was given and also the angle A, and we want to find the side a. Write down a and underneath it put b, thus, ; on referring to the figure we see that this fraction represents the trigonometrical ratio sin. A. We now put the fraction equal to the trigonometrical ratio thus, = sin. A, therefore, a = 6 x sin. A. Hence the side a will be found by adding together the logarithm of b and the sin. A, rejecting to from the index and taking out the natural number corresponding to the sum of the logarithms. The other side of the triangle is found in a similar manner. To find the remaining acute angle, subtract the given angle from b, g 90° GENERAL RULE TO FIND AN ANGLE WHEN TWO SIDES ARE GIVEN.-In (2) to solve the triangle, we have to find the two acute angles and the remaining side. When an angle is not given we must always first find one of the angles. To do this, write down in the form of a fraction the two given sides (it does not matter which is put on top), put the corresponding trigonometrical ratio equal to this fraction, and the formula is complete for finding the angle. To exemplify this, suppose in the Fig. 2, p. 44, a and b are given, and we want to find the angle A. Form a fraction with a and b thus, î; referring to the figure we see that this fraction represents the trigonometrical ratio sin. A. We now put this trigonometrical ratio equal to the fraction, thus, sin. A Hence the angle A will be found by subtract b ing the logarithm of b from the logarithm of a plus 10, and the degrees i on a |