Let x = the number of degrees required. Then x is subtended by an arc of the length of the radius = r. Now an angle of 180° is subtended by a semicircle; i.e. by an arc = tr, where a = 3:14159. It is, however, generally more convenient, and for our present purposes necessary, to determine an angle not by an arc, or by the ratio of an arc to its radius, but by certain straight lines, or by the ratio of certain straight lines to each other. These lines, or, as they are now more commonly regarded, these ratios, are called respectively the sine, tangent, secant, cosine, cotangent, or cosecant of the angle. We proceed to define these terms. Let AOB be an angle A. Draw OC perpendicular to 04, and with the centre O and any radius OA describe an arc of a circle, meeting OC in C. Draw Bn, Bm, perpendicular to 0A, OC; at A and C draw At, Cu, perpendicular to OA, OC. Then Bn is defined to be the sine of the angle AOB, to the radius OA, B m n A At is defined as the tangent of AOB. secant of AOB. versed sine of AOB. Hence, also Bm (or On) is the sine of BOC. tangent of BOC. secant of BOC. Now BOC is 90° – A. And Bm, Cu, Ou, are defined as being the cosine, cotangent, cosecant respectively of AOB. Hence Bn = sine A to the radius OA. To this method there is the obvious objection that the sines, J&c., of a given angle have different values, according as they are referred to different radii, accordingly, instead of defining the sines, &c., as lines, it is, as above stated, now more usual to define them as ratios; by which means all consideration of the radius to which the sines are referred is avoided. According to this method the definitions are given as follow : CA B any right angled triangle having the right angle at C. N.B.—The angle which with another makes up 90° is called the complement of that angle. Hence B is the complement of A; and the cosine, cotangent, and cosecant of an angle are evidently the sine, tangent, and secant of its complement. It is plain (Euc. VI. 4) that the values of these ratios depend solely on the angle, and are quite independent of the magnitude of the sides of the triangles. If, then, we can by any means calculate the value of these ratios, which correspond to any angle, these values can be arranged in a table; and it is plain that, having such tables, if we have given any one of the ratios defined above, we know the angle; and vice versa, if we have the angle given, we know the ratio. Such tables have been calculated on principles to be hereafter explained ;-for our present purpose it is sufficient for us distinctly to understand, that if we have given the numerical value of any one of the ratios, we know the angle that corresponds to it, and vice 5. On the relations between the Trigonometric Ratios of the same Angle. Let A B C be a right-angled triangle, It is of very great importance that the student be familiar with the relations we have just established. He will therefore do well to perform the following exercises : (8). Express each of the trigonometrical ratios of an angle A in terms of the sine of A. There are two or three angles, the numerical values of the trigonometrial ratios of which can be easily determined. These angles are 45°, 60°, and 30°. 6. To find the Trigonometrical Ratios of an Angle of 45°. A B C, a right angled triangle. C the right angle. 7. To find the Trigonometrical Ratios of an Angle of 60°. A B C, an equilateral triangle. The angle A B C is one of 60°. Draw AD per pendicular to B C. Now BD = 1. BC = 4. AB and A D2 = A B2 — BD2 = 4. A B2. Since 30o = 90' 60° we shall have sin. 30°= cos. 60° 2 And similarly tan. 30o = sec. 30o = V 3 8. Gencralization and Extension of the Principles and Definitions previously laid down. The definitions above given hold good for angles that are less than ninety degrees; the definition, both of an angle and of the ratios which determine it, admit of and require extension; the nature of which extension and the principle on which it is made we will now proceed to explain. 9. The use of the Negative Sign to denote position. с AL JB Let A B be a line, the length of which is a. Let BC be a line, the length of which is b. Then it is plain that A C is a — -6. This distance, A C, is arrived at by measuring a distance (a) to the right from A, and then measuring another distance (6) to the left from B, the ta and the — b being measured in opposite directions. It appears then that when a stands for a line measured from a given point in one direction, - a will stand for a line of the same length measured in the opposite direction. In other words, the magnitude of the line is determined by the number of units in Ag while the direction is determined by its sign. It is generally understood that + a signifies a line measured to the right of a given point, as AB, and therefore that signifies a line À measure to the left of the fixed point, as A B’. a - a 2 10. Extension of the Definition of an Angle. a P We now proceed to extend the definition of an angle. An angle, as defined by Euclid,—i. e. as the inclination of one line to another,-must be less than two right angles. But if we regard an angle as the space swept out by a right line revolving in one plane, about a fixed point in a given straight line, we clearly remove the limit imposed by Euclid's definition on the magnitude of the angle. Thus if A be the fixed point in the fixed line AB, AP the moveable line, let the angle BAP, according to Euclid's definition, be A. Now it is plain that in one revolution A P passes through an angle equal to four right angles, or 360°. Á Morcover, A P will always come to its present position after one, two, or any number of revolutions; and therefore, according to our extended definition, BAP may be either A or 360° + A, or 2 X 360° + A, or, generally, 360° n + A, where n is integer. B n |