SINE

(Lat. sinus, Me bosom.) In Trigonometry, the sine of any arc of a circle is the straight line drawn from one extremity of the arc perpendicular to the radius pa ing through the other extremity. Thus, in the circle A B C D, let O A and O B be two radii, and let B X be perpendicular to O A; then B X is the sine of the intercepted arc A B. In like manner, if C Y and D Z be perpendicular to A O, or its prolongation O A'; then C Y IS the sine of the arc A B C, or of A' C, the supplement of ABC, and D Z the sine of A CD, or of A' D. Draw the diameter P O P', and suppose the arc A B to increase from 0 to 360°. It is evident that as the arc A B increases from A to A P, or from 0 to 90°, the sine B X increases from zero to O P, where it becomes equal to the radius; and hence the radius is called the sinus tot us, or whole sine. Between P and A the sine again diminishes as the arc is increased, and vanishes at A'. From A towards P' it again increases till it becomes equal to the radius O P', and from this position decreases until it again vanishes at A. Hence it appears that the origin being supposed at A, the sine is positive in the first and second quadrants, or from 0 to 180°; and negative in the third and fourth, or from 180° to 360°. If we conceive another circle to be described about the same centre O, the two lines O A and O B would intercept on it an arc similar to A B; and if, also, we suppose the radius of this second circle = 1, and denote the inter  [s. 1132]