= 1 cos. (90°-8) sin. by the last case, ¡, or, cosec. 8 sin. =1 (6) Since CAT is a right angled triangle, and CT the hypothe The above results, which are of the highest importance in all trigonometrical investigations, are collected and arranged in the following table, which ought to be committed to memory: 12. The chord of an arc is the ratio of the straight line joining the two extremities of the arc to the radius of the circle. PROPOSITION. The chord of any arc is equal to twice the sine of half the arc. Take any arc AQ, subtending at the centre of the circle the angle ACQ=0. Draw the straight line CP bisecting the angle ACQ. Join A, Q; from P let fall PS dicular on CA. perpen Since CP bisects ACQ, the vertical angle of the isosceles triangle ACQ, it bisects the base AQ at right angles. B S C Ꭰ E Therefore, AO=OQ, and the angles at O are right angles. Again, since the triangles AOC, PSC, have the angles CSP, COA, right angles, and the angle PCS common to the two triangles, and also the side CP of the one equal the side CA of the other, these triangles are in every respect equal. We shall now proceed to explain the principle by which the signs of the trigonometrical quantities are regulated All lines measured from the point C along CA, that is, to the left, are considered positive, or have the sine +. All lines measured from the point C along CD, that is, in the opposite direction A to the right are considered negative, or have the sign All lines measured from the point C along CB, that is, upwards, are considered positive or have sign +. All lines measured from the point C along CE, that is, in the opposite direction downwards, are considered negative, or have the sign Let us determine according to this principle, the signs of the sines and cosines of angles in the different quadrants. Here PS Cc is reckoned from C along CB upwards, and is therefore positive. CS is reckoned from C along CA, to the left and is therefore positive. In the first quadrant, therefore the sine and cosin are both positive. In the second quadrant, sin. = P, S, 2 2 CA Here P,S,=Cc, is reckoned from C along CB upwards, and is therefore positive. CS, is reckoned from C along CD to the right and is therefore negative. 8/ A D In the second quadrant, therefore, the sine is positive and the cosine negative. Here P,S,=Cc, is reckoned from C along CE, downwards, and is therefore negative. CS, is reckoned from C along CD, to the right, and is therefore negative. Here P,S,=Cc, is reckoned from C along A S E D In the fourth quadrant, therefore, the sine is negative, and the cosine positive. Hence we conclude, that the sine is positive in the first and second quadrants, and negative in the third and fourth; and the cosine is positive in the first and fourth, and negative in the second and third, or in other words: The sine of an angle less than 180° is positive and the sine of an angle greater than 180° and less than 360° is negative. The cosine of an angle less than 90° is positive, the cosine of an angle greater than 90°, and less than 270°, is negative, and the cosine of an angle greater than 270°, and less than 360,° is positive. The signs of the sine and the cosine being determined, the signs of all the other trigonometrical quantities may be at once established by referring to the relations in Table 1. Thus, for the tangent, tan. = sin. cos. 8 Hence, it appears that when the sine and cosine have the same sign the tangent will be positive, and when they have different signs it will be negative. Therefore, the tangent is positive in the first and third quadrants and negative in the second and fourth. The same holds good for the cotangent; for the sign of the secant is always the same with that of the cosine; and, since cosec. = 1 sin. in like manner, the sign of the cosecant is always the same with that of the sine. The versed sine is always positive, being reckoned from A always in the same direction. P B C It is sometimes convenient to give different signs to angles themselves. We have hitherto supposed angles of different magnitudes to be generated by the revolution of A the moveable radius CP round C in a direction from left to right; and the angles so formed have been considered positive, or affected with the sign +. If we now suppose the angle = to be generated by the revolution of the radius CP' in the opposite direction, we may, upon a principle analogous to the former, consider the angle as negative, and affect it with the sign —. P' We shall now determine the variations in the magnitude of the sine and cosine for angles of different magnitudes. In the first quadrant : Let CP, CP,, CP,,... be different positions of the revolving radius in the first quadrant; and from P, P,, P. draw PS, P,S,, P,S,, perpendiculars on CA. 39 When the angle becomes very small, PS becomes very small also; and when the revolving radius coincides with CA, that is, when the angle becomes 0, then PS disappears altogether, and is =0. On the other hand, when the angle becomes equal to 90°, PS coincides with CB, and is equal to it. PS Hence since, generally, sin. =. and since, when =90°, PS=CB; CA' CB .. sin. 90° CA =1; .'. CB=CA. Again, it is manifest, that as the angle increases the cosine diminishes; for and C2>CA СА |