77. The following diagrams shew the signs of the trigonometrical functions in the four quadrants. It will be sufficient to consider the three principal functions only. The diagram below exhibits the same results in another useful form. 78. When an angle is increased or diminished by any multiple of four right angles, the radius vector is brought back again into the same position after one or more revolutions. There are thus an infinite number of angles which have the same boundary line. Such angles are called coterminal angles. If n is any integer, all the angles coterminal with A may be represented by n. 360°+4. Similarly, in radian measure all the angles coterminal with 0 may be represented by 2nñ+0. From the definitions of Art. 75, we see that the position of the boundary line is alone sufficient to determine the trigonometrical ratios of the angle; hence all coterminal angles have the same trigonometrical ratios. Example. Draw the boundary lines of the angles 780°, 130°, - 400°, and in each case state which of the trigonometrical functions are negative. (1) Since 780=(2 × 360)+60, the radius vector has to make two complete revolutions and then turn through 60°. Thus the boundary line is in the first quadrant, so that all the functions are positive. (2) Here the radius vector has to revolve through 130° in the negative direction. The boundary line is thus in the third quadrant, and since OM and MP are negative, the sine, cosine, cosecant, and secant are negative. M (3) Since -400 (360+40), the radius vector has to make one complete revolution in the negative direction and then turn through 40°. The boundary line is thus in the fourth quadrant, and since MP is negative, the sine, tangent, cosecant, and cotangent are negative. EXAMPLES. VIII. a. State the quadrant in which the radius vector lies after describing the following angles : For each of the following angles state which of the three principal trigonometrical functions are positive. 15. 16. 17. 13π In each of the following cases write down the smallest positive coterminal angle, and the value of the expression. 79. Since the definitions of the functions given in Art. 75 are applicable to angles of any magnitude, positive or negative, it follows that all relations derived from these definitions must be true universally. Thus we shall find that the fundamental formulæ given in Årt. 29 hold in all cases; that is, cos A × sec A = 1, tan A× cot A=1; sin A x cosec A=1, It will be useful practice for the student to test the truth of these formulæ for different positions of the boundary line of the angle A. We shall give one illustration. It thus appears that the truth of these relations depends only on the statement OP2=MP2+OM2 in the right-angled triangle OMP, and this will be the case in whatever quadrant OP lies. NOTE. OM2 is positive, although the line OM in the figure is negative. 81. In the statement cos A=√1-sin2 A, either the positive or the negative sign may be placed before the radical. The sign of the radical hitherto has always been taken positively, because we have restricted ourselves to the consideration of acute angles. It will sometimes be necessary to examine which sign must be taken before the radical in any particular case. Since sin2 4 + cos24=1 for angles of any magnitude, we have sin A = √1-cos2 A. Denote 126° 53' by A; then the boundary line of A lies in the second quadrant, and therefore sin A is positive. Hence the sign + must be placed before the radical; The same results may also be obtained by the method used in the following example. The appropriate signs of the lines are shewn in the figure. |