Let r denote the radius, and -3.14159265358979... ra semi-circumference = 180° = 200′ = two right angles. r = a quadrant 90° 100' one right angle. 2 r = a circumference angles. If r 1, the above expressions become, respectively, 2π. 33. Unit of Circular Measure. The unit of circular measure is that angle at the center whose intercepted arc is equal in length to the radius. Let u denote the unit of circular measure, and r the radius. Then, since the semi-circumference, u = 180° = 2001. Let d, g, c, respectively, denote the number of degrees, grades, and units of circular measure in an angle; then, 34. Origin, Termini and Situation of Arcs. The origin of an arc is the extremity at which it begins. The primary origin of arcs is at the right extremity of the primary diameter. The secondary origin of arcs is at the upper extremity of the vertical diameter. S. N. 3. The terminus of an arc is the extremity at which it ends. P P T T" An arc is said to be situated in that quadrant in which its terminus is situated, thus: The arc OT is in the first quad rant. The arc 00'T' is in the second quadrant. The arc OPT" is in the third quadrant. 35. Positive and Negative Arcs. Positive arcs are those which are estimated in the direction contrary to that of the motion of the hands of a watch. Negative arcs are those which are estimated in the same direction as that of the motion of the hands of a watch. Thus, OT, OT, OT", OT"", estimated to the left, are positive, and OT"", OT", OT', OT, estimated to the right, are negative. 36. The Complement of an Arc. The complement of an arc or angle is 90° minus that arc or angle. If the arc or angle is less than 90°, its complement is positive. If the arc or angle is greater than 90°, its complement is negative. The complement of an arc, geometrically considered, is the arc estimated from the terminus of the given arc to the secondary origin. Therefore, by the preceding article, the complement of an arc will be positive or negative, according as the arc is less or greater than 90°. TO' is the complement of OT, and is positive. T'O' is the complement of OT', and is negative. 37. The Supplement of an Arc. The supplement of an arc or angle is 180° minus that arc or angle. If the arc or angle is less than 180°, its supplement is positive. If the arc or angle is greater than 180°, its supplement is negative. The supplement of an arc, geometrically considered, is the arc estimated from the terminus of the given arc to the left-hand extremity of the primary diameter. Therefore, by article 35, the supplement of an arc will be positive or negative, according as the arc is less or greater than 180°. TP is the supplement of OT, and is positive. TRIGONOMETRICAL FUNCTIONS. 38. Preliminary Definitions and Remarks. 1. A function of a quantity is a quantity whose value depends on the given quantity. 2. The trigonometrical functions, called also circular functions, are auxiliary lines, which are functions of an arc or of the angle which has the same measure as that arc. 3. These functions are eight in number, and are called the sine, co-sine, versed-sine, co-versed-sine, tangent, co-tangent, secant and co-secant, which are abbreviated thus, sin, cos, vers, covers, tan, cot, sec, cosec. 4. The solution of triangles is accomplished by the aid of these functions, since they enable us to ascertain the relations which exist between the sides and angles of triangles. 5. The primary origin will be taken as the common origin of the arcs, unless the contrary is stated. 6. The origin of any arc, wherever situated, may be considered the primary origin of that arc; and its secondary origin is a quadrant's distance from the primary origin, in the direction of the positive or negative arcs, according as the given arc is positive or negative. 7. An arc will be considered positive unless the contrary is stated. 8. The primary diameter passes through the primary origin; and the secondary diameter, through the secondary origin. 9. Lines estimated upward, toward the right, or from the center toward the terminus of the arc, are considered positive. 10. Lines estimated downward, toward the left, or from the center and the terminus of the arc, are considered negative. 11. The limiting values of the circular functions are their values for the arcs 0°, 90°, 180°, 270°, 360°. 12. The sign of a varying quantity, up to a limit, is its sign at the limit. 13. Point out positive arcs in the following diagram, and the origin and terminus of each. 14. Point out negative arcs, the origin, terminus and primary diameter of each. 15. Point out the positive lines, also the negative. 39. The Sine of an Arc. The sine of an arc is the perpendicular distance of its terminus from the primary diameter. MT is the sine of the arc OT. M'T' is the sine of the arc OT'. M'T" is the sine of the arc OT". MT"" is the sine of the arc OT". By the arcs OT" and OT", we are to understand the positive arcs, and not the negative arcs designated by the same letters. T T The sine of an arc is the sine of the angle measured by that are. Thus, MT, the sine of the arc OT, is the sine of the angle OCT, which is measured by the arc OT; and similarly for the other arcs and angles. The arcs OT and OT' are in the first and second quadrants, respectively, and their sines MT and M'T' are estimated upward, and are therefore positive; hence, The sine of an arc in the first or second quadrant is positive. The arcs OT" and OT" are in the third and fourth quadrants, respectively, and their sines, M'T" and MT"", are estimated downward, and are therefore negative; hence, The sine of an arc in the third or fourth quadrant is negative. Let the chord TT' be parallel to the primary diameter OP, then will M'T' be equal to MT, and the arc OT will be equal to the arc T'P; but the arc T'P is the supplement of the arc OT'; therefore, the arc OT is the supplement of the arc OT'; but M'T', |