Let OP1, OP, OP, OP, represent the position of the revolving line at any period of revolution in the several quadrants respectively, And let PIN, PN2, PN, PN4 be the respective perpendiculars from the end of the revolving line upon the initial line. Then P,N, PN2, PN, PN4 are respectively the perpendiculars corresponding to the angles generated. 4 Also, ON, ON, ON, ON, are respectively the bases of the right-angled triangles with respect to the angles in question. We have then in the second quadrant— P,N, Sin AOP, = 2, cos AOP2 tan AOP,, = P2N2, &c. ON2 = ON It is therefore evident that the relations between the trigonometrical ratios, which were proved to exist in Art. 7, also hold for angles in the second quadrant—that is, angles between 90° and 180°. And in the same way we may show that they hold for angles in the third, fourth, or any quadrant. And again, if we suppose the line to revolve in a negative direction, and take the position OP', we shall have P'N' the perpendicular corresponding to the negative angle AOP', and ON' the base. And the relations proved in Art. 7 may be also similarly proved to exist here. Hence the relations proved in Art. 7 hold for any angles whatever. Changes of Magnitude and Sign of the Trigonometrical Ratios. 11. Let OP, OP, OP, OP, be positions of the revolving line in the several quadrants respectively; PN1, P2NË PN, PN4, the respective perpendiculars; and ON1, ON 2, ON, ON, the bases of the corresponding right-angled At the commencement of the motion of the revolving line, the angle AOP1 = 0°; Also, the perpendicular P1N1 And the base ON Hence, we have = OP1 tan 0° 0. As the revolving line moves from OA towards OB, P1N1 increases and ON, diminishes; and when it arrives at OB, we have PIN OP,,and ON1 = 0. But the angle generated is now a right angle. Hence we have Hence, as the angle increases from 0° to 90° The sine changes in magnitude from 0 to 1 and is +. Here the perpendicular P,N, is +, and the base ON2 is *The student ought properly to look upon the values 0, 1, 0 here obtained as the limiting values of the sine, cosine, and tangent respectively, when the angle is indefinitely diminished. Hence the sine during the second quadrant is +, the cosine is and the tangent is 2 Again, as the revolving line moves from OB to OA', the perpendicular P2N, diminishes until it becomes zero. Also, the base ON, increases in magnitude, until it finally coincides with OA', and .. OP2. But the angle now described is 180°. Hence we have = Hence in the second quadrant The sine changes in magnitude from 1 to 0, and is positive. (3.) In the third quadrant— The sine changes in magnitude from 0 to 1, and is negative. The cosine changes in magnitude from 1 to 0, and is negative. The tangent changes in magnitude from 0 to ∞, and is positive. (4.) In the fourth quadrant The sine changes in magnitude from 1 to 0, and is negative. The cosine changes in magnitude from 0 to 1, and is positive. The tangent changes in magnitude from ∞ to 0, and is negative. Moreover, as the cosecant, secant, and cotangent are respectively the reciprocals of the sine, cosine, and tangent, it follows that their signs will follow respectively the latter, and that their magnitudes will be their reciprocals. CHAPTER IV. TRIGONOMETRICAL RATIOS CONTINUED. ARITHMETICAL VALUES OF THE TRIGONOMETRICAL RATIOS OF 30°, 45°, 60°, &c. |