15. Functions of Complemental Angles. - In the rt. A ABC we have Therefore the sine, tangent, secant, and versed sine of an angle are equal respectively to the cosine, cotangent, cosecant, and coversed sine of the complement of the angle. 16. Representation of the Trigonometric Functions by Straight Lines. — The Trigonometric functions were formerly defined as being certain straight lines geometrically connected with the arc subtending the angle at the centre of a circle of given radius. Thus, let AP be the arc of a circle subtending the angle AOP at the centre. Draw the tangents AT, BT' meeting )P produced to T', and draw PC, PD I to OA, OB. Since any arc is the measure of the angle at the centre which the arc subtends (Art. 5), the above functions of the arc AP are also functions of the angle AOP. It should be noticed that the old functions of the arc above given, when divided by the radius of the circle, become the modern functions of the angle which the arc subtends at the centre. If, therefore, the radius be taken as unity, the old functions of the arc AP become the modern functions of the angle AOP. Thus, representing the arc AP, or the angle AOP by 0, we have, when OA = OP=1, and similarly for the other functions. Therefore, in a circle whose radius is unity, the Trigonometric functions of an arc, or of the angle at the centre measured by that arc, may be defined as follows: The sine is the perpendicular let fall from one extremity of the arc upon the diameter passing through the other extremity. The cosine is the distance from the centre of the circle to the foot of the sine. The tangent is the line which touches one extremity of the arc and is terminated by the diameter produced passing through the other extremity. The secant is the portion of the diameter produced through one extremity of the arc which is intercepted between the centre and the tangent at the other extremity. The versed sine is the part of the diameter intercepted between the beginning of the arc and the foot of the sine. Since the lines PD or OC, BT', OT, and BD are respectively the sine, tangent, secant, and versed sine of the arc BP, which (Art. 12) is the complement of AP, we see that the cosine, the cotangent, the cosecant, and the coversed sine of an arc are respectively the sine, the tangent, the secant, and the versed sine of its complement. EXAMPLES. = sec A. 66 66 1. Prove tan A sin A +cos A 2. cot Acos A sin A = cosec A. 3. (tan A - sin A)?+ (1 - cos A)'=(se: A – 1)? 4. tan A +cot A = sec A cosec A. 5. (sin A +cos A) = (sec A + cosec A)=sin A cos A. 6. (1+tan A)'+(1+cot A)=(sec A + cosec A). 66 = 66 7. Given tan A cot 2 A; find A. sin A = cos 3 A; find A. sin A = cos (45° - 4A); find A. 10. tan A= cot 6 A; find A. 11. cot A =tan(45° + A); find A. = B P M' 17. Positive and Negative Lines. - Let AA' and BB' be two perpendicular right lines intersecting at the point 0. Then the position of any point in the line AA' or BB' will be determined if we know the distance of the point from 0, and if we know also upon which side of O the Á А. 0 м point lies. It is therefore convenient to employ the algebraic signs + and so that if dis P' B tances measured along the fixed line OA or OB from 0 in one direction. be considered positive, distances measured along OA' or OB' in the opposite direction from O will be considered negative. This convention, as it is called, is extended to lines parallel to AA and BB'; and it is customary to consider distances measured from BB' towards the right and from AA' upwards as positive, and consequently distances measured from BB' towards the left and from AA' downwards as negative. 18. Trigonometric Functions of Angles of Any Magnitude. — In the definitions of the trigonometric functions given in Art. 13 we considered only acute angles, i.e., angles in the first quadrant (Art. 5), since the angle was assumed to be one of the acute angles of a right triangle. We shall now show that these definitions apply to angles of any magnitude, and that the functions vary in sign according to the quadrant in which the angle happens to be. I. When A lies in the 1st quadrant, MP is positive because measured from upwards, OM is positive because measured from 0 towards the right (Art. 17), and OP is positive. Hence in the first quadrant all the functions are positive. II. When A lies in the 2d quadrant, as MP OP and therefore sec A and cot A are negative, and cosec A is positive (Art. 13). |