Hence we conclude that if A be any geometrical angle, its most general trigonometrical form will be. n × 360° + A, where n is any positive or negative integer whatever. 12. Extension of Definitions of Trigonometrical Ratios. We now proceed to consider the trigonometrical ratios of angles greater than a right angle. We shall, in the first instance, confine our attention to the sines and cosines of angles, less than four right angles. We have already explained that Now as AN is measured along AB to the right, AN is positive. And if we reckon lines measured upward, from A towards C positive, it is plain that PN, being measured parallel to that direction, is positive. The signs of the sine and cosine of an angle less than ninety degrees are then, by this way of reckoning the signs of the measurements, positive, as they should be. Now if we consider an angle BAP1, it is clear that P, N' stands in the same relation to BAP, that PN does to BAP. It is plain that P, N' is positive, and A N' is negative. Hence is negative. 180° is positive, and the cosine of an angle 7 90° 180° In like manner, if B A P2 be the angle subtended by the circumference B P1 P2 1 Now P2 N' is negative, and A N' is negative. Hence the sine of an angle 7180' 270° is negative, and the cosine of an angle 7180°/270° is negative. In like manner if PA B signify the angle subtended by the circumference B P1 P2 P3, 1 2 AN AP3' And P, N is negative, and A N is positive. Hence, sine of an angle 7270° ≤360° is negative and the cosine of an angle 7270° 360 is positive. These four angles which we have considered are said to be in the first, second, third, and fourth quadrants respectively. By means of the above, if we have given the signs both of sine and cosine of an angle, we can tell in what quadrant it must lie. n, must lie in the second quadrant; i. e. (13.) To express the Trigonometrical Ratios of any angle in terms of those of an angle less than 90°. Again, the trigonometrical ratios of any angle can be expressed by means of the ratios of an angle less than 90°. 29 For if, in the same figure, BAP, B'A P1, B'A P2, BAP, are equal to one another, and therefore the lines P N, P, N', P, N', P, N, are equal in' magnitude; as also are AN and AN. If then we take account both of sign and magnitude, sin. (360° 45°) = If consider the case of the tangent of an angle, LA being an angle, ≤90°. In the same manner we may express the other trigonometrical ratios of angles greater than 90° by means of those angles less than 90. It is to be observed that if we suppose AP to make one complete revolution from A P, it returns to its present position. So that AN and N P are the same both in magnitude and direction for the angle 360° + A as for A. The same is true of any number of complete revolutions. Hence, if ƒ denote any trigonometrical ratio whatever, There are a great variety of relation similar to those above deduced. The following are worth notice :— This result can be arrived at by reference to formulas previously proved. We have seen that under all circumstances f(n.360°+A) = ƒ (A). .*. sin. (360° — A) = sin. (— A). 14. On the Magnitudes of the Trigonometrical Functions of Angles, 0°, 90°, 180°, 270°. Now if A P coincide with A B, A=0 and P N=0, .. sin. 0° = 0. If A P revolve round A, P N increases until A P coincides with A C, when A = 90°, and PNAP. .. sin. 90° 1. After which, as AP revolves towards A B', PN decreases until at AB coincides with A B1, when PN0, A180°, and therefore Sin. 110° 0. As AP revolves from A B' towards A C, P N increases negatively until AP coincides with A C, when A= 270° and PN=- A P. .. sin. 270° - — 1. As AP revolves from A C' towards its original position A B, PN decreases negatively, until when A P coincides with A B, we have A = 360°, and P N vanishes. .. sin. 360° = 0. In like manner, if we take the cosine we have If we apply similar reasoning to the various trigonometric functions that employed in discussing the variations of the sine of A, we obtain results which may be arranged in a tabular form, as follows: The student will do well to verify carefully all the results given in this table; he will also observe that the trigonometrical ratios illustrate the principle that if a function of a variable changes its sign, it must pass through the values of either zero (.0) or infinity (∞). As the values of the ratios are continuous, the ratios increase gradually to their greatest value, and then decrease to their least. Thus, to take the case of the sine of an angle, which we call 0. Sine increases from 0, when 0 — 0° up to 1, when 90°. It then decreases to 0, when 0=180°; after which it still further decreases till it equals 1, when 0=270°, and finally increases up to 0, when = 360°. 15. To determine all the angles which have the same sine, or cosine, &c. There is another class of questions presented to us by this extension of our definition of an angle, viz., having given a trigonometrical ratio of an angle, to find all the angles corresponding to it. For example, tan. 0 = p. Now, if we did not reckon any angles but those less than 180°, as is the case in geometry, we could only have one value of corresponding to a given value of tan. 0. Suppose this value = a. Then, if we take the trigonometrical or generalized conception of an angle, we shall have another 180°+a. And since no trigonometrical ratio changes either its value or its sign when its angle is increased by any multiple of 360°, it is plain that in addition to a we shall have a series of values, 360°+a, 2 × 360°+ a, 3 × 360° + a + . ,n 360° a; and in addition to the value 180° + a, we shall have a series of values, 360° + 180° + a, 2 × 360° + 180° + a, 3 × 360° + 180°+a,.... n 360° +180° + a. Both these series may be included in one formula, . em 180° +a, where m is any integer number whatever. In the same manner, if Cos. 0=4, and a is the value of less than 180°, which has for its cosine 7, then all the values of which have a cosine q are included in the formula, |