35. If the terminal line of an angle lies between OX and OY, the angle is said to be in the first quadrant; if between OY and OX', in the second quadrant; between OX' and OY', in the third quadrant; between OY' and OX, in the fourth quadrant. Thus, any positive angle between 0° and 90°, or between 360° and 450°, or any negative angle between - 270° and - 360°, is in the first quadrant. Any positive angle between 90° and 180°, or between 450° and 540°, or any negative angle between 180° and - 270°, is in the second quadrant. 36. We observe, by inspection of the results in Art. 33, the following points in regard to the algebraic signs of the trigonometric functions in the different quadrants: For any angle in the first quadrant all the functions are positive. In the second quadrant, the sine and cosecant are positive, and the cosine, tangent, cotangent, and secant are negative. In the third quadrant, the tangent and cotangent are positive, and the sine, cosine, secant, and cosecant are negative. In the fourth quadrant, the cosine and secant are positive, and the sine, tangent, cotangent, and cosecant are negative. 37. It is customary to express the foregoing principles in tabular form, as follows: FUNCTIONS OF 0°, 90°, 180°, 270°, AND 360°. 38. To find the functions of 0° and 360°. The terminal line of 0° coincides with the initial line OX. Let P be a point on OX, such that OP-a. Then by Art. 31, the co-ordinates of P are (a, 0). By Art. 34, the functions of 360° are the same as those of 0°. Let P be a point on OY, such that OP= a. Let P be a point on OX', such that OP= a. Let P be a point on OY', such that OP= a. Then the co-ordinates of P are (0, − a). Whence by definition, == Note. No absolute meaning can be attached to such results as cot 0° = ∞, tan 90° = ∞, etc. The equation cot 0° merely signifies that as an angle approaches 0° as a limit, its cotangent increases without limit. A similar interpretation must be given to the equations csc 0° = ∞, sec 90° =∞, etc. FUNCTIONS OF (-A) IN TERMS OF THOSE OF A. There will be four cases, according as A is in the first, second, third, or fourth quadrant. In each figure, let the positive angle XOP (indicated by the full arc) represent the angle A, and the negative angle XOP' (indicated by the dotted arc) the angle (-A). Draw PP' perpendicular to XX'; then the right triangles OPM and OP'M have the side OM and the angle POM of one equal to the side OM and the angle P'OM of the other, and are equal. Whence, PM= P'M and OP = OP'. Let PMP'M= a, OM=b, and OP = OP' = c. CASE I. A in the first quadrant, — A in the fourth. It is evident from the above that the cosine and secant of -A are equal respectively to the cosine and secant of A, while the sine, tangent, cotangent, and cosecant of — A are equal respectively to the negatives of the sine, tangent, cotangent, and cosecant of A. Whence we obtain the formulæ (9). Whence we obtain the formulæ (9) as before. |