40. Congruent Angles. If the difference of two angles, a and B, is n times 360° (where n is one of the numbers 0, 1, 2, 3, etc.), they are said to be congruent angles and we write aß; read: a is congruent to B. Thus 15° 375°, — 172° ... 188°, etc. If two angles y and 8 are not congruent, they are said to be incongruent, and we write y 8, 8y. Thus, 45° 400°. To prove that two angles are congruent it is necessary and sufficient to show that their difference is either O or a multiple of 360°; that is, if a≈ ß, a− ß = ± n ⋅ 360°; and conversely. 41. Properties of Congruent Angles. If two congruent angles are placed on the same pair of axes, their terminal sides will coincide. For example, any two of the angles 50°, 410°, - 310° are congruent; when placed on the same axes their terminal sides all coincide. If two incongruent angles be placed on the same axes, their terminal sides will not coincide.* This is the geometric equivalent of § 40. (1) Every angle obtained by putting n = 0, 1, 2, 3, etc. in the formula a±n 360° is congruent to a; conversely every angle congruent to a is found in this set. (Use § 40.) (2) If a is any angle whatever, there is one and only one angle between 0° and 360° (0° included, 360° excluded) which is congruent to α. For if a is an angle of any size, the addition to a and subtraction from a of successive multiples of 360° (360°, 720°, 1080°, etc.) will give all angles congruent to a, and obviously one and only one of these lies between 0° and 360°. (3) If ay and if ẞy, then a ß. That is, if each of two angles is congruent to a third angle they are congruent to each other. Proof: α- γ = ±m • 360°, and whence, adding: a− ẞ=(±m±n) 360°, that is, a≈ß. γ β 360°; *The word congruent is thus equivalent to the word superposable as used in geometry; but we must remember that two angles are superposable if and only if it is possible to make them coincide vertex with vertex, initial side with initial side, terminal side with terminal side. That such angles are not identical is evident in such practical instances as rotating machinery; the motion of a flywheel, 30° per second, differs essentially from that of a wheel turning 410° (or from that of one turning 310° per second). See § 37, p. 51. (4) If the same angle be added to (or subtracted from) each of two congruent angles, the results will be congruent angles. β Given aẞ, to prove: (a) a + y =ẞ + y, (b) α-y=By. (a) (x+y)-(B + y) = α − ß = ±n 360°; hence a+y=B+y. (b) (α-y)-(B− y) = α-ẞ= n.360°; hence a-yẞ-y. (5) The negatives of congruent angles are congruent. Given aẞ, to prove that (a)(B). Proof: (B)-(—α)=α-ẞ= ±n 360°; hence (B)=(-a). By (4) the transposition of a term from one side of a congruence to the other with change of sign is permissible; e.g. from 45° - 350° ≈ 55° follows 45° 55° + 350°; from a + 150° ≈ ß + 180° follows a B 30°. By (5) it is permissible to change the signs of all terms of a congruence; e.g. from 2 x 18°3 x 63° follows x45° and x 45°. It is not ordinarily permissible to multiply or divide both sides of a congruence by any number (except to multiply by an integer); e.g. from 30° 390° it does not follow that 10° 130°. EXERCISES XIX. — CONGRUENT ANGLES 1. Draw figures on polar coördinate paper to illustrate (4), § 41, when (a) α = 30°, B = 390, y = 20°; (b) α = 90°, ß == 270°, y = 45°; (c) α = 72°, B = 432°, y = 72°. 2. Taking α = 60°, B = 300°, y = 50°, d = 310°, draw a figure showing that (a) α ≈ ß; (b) y≈d; (c) α — y ≈ ß — d. 3. Find the angle between 0° and 360° which is congruent to each of the following: (a) - 42° 13'; (b) — 842°; (c) 364° 23'; (d) 360°; (e) — 90°; (ƒ) 420°; (g) 2700°. x360° + 2 x. Ans. x = 9°, i.e. x = = 9° ± n · 360°. 5. Find 3 values for x which satisfy the congruence 3x-70°~150°—x. Ans. x=- - 35°, 55°, 145°. 6. Find the smallest positive value of x which satisfies the congruence : x+200° 40° - 3x. Ans. x = = 50°. 7. Prove that the sum of the interior angles of a convex polygon is congruent to 0° or 180° according as the number of sides is even or odd. 8. Compare a rotational speed of 30° per sec. with a speed of 390° per sec. 9. Reduce an angular speed of 390° per second to revolutions per second; to revolutions per minute. 10. If aẞ and y≈ 8, prove that a + y≈ ß + 8, and α — y ≈ ß — d. Compare (4), § 41. PART II. ANGULAR SPEED-RADIAN MEASURE 42. Measurement of Angles. An angle may be named and used before it is expressed in any system of measurement. Thus, we may refer to an angle A of a right triangle whose perpendicular sides are 16 in., and 24 in., respectively; and we can compute tan A-24/16 1.5, etc., without measuring A in terms of any unit angle. General theorems like the law of sines remain true in any system of measurement. The measure of an angle — say 36° — consists of two distinct ideas: the unit angle (in this example, one degree) and the abstract number (here 36) which expresses the numerical measure of the angle in terms of the chosen unit. The elementary units are defined in § 37. For many purposes it is convenient to use another unit angle called the radian. 43. Radian Measure of Angles.* A radian is a positive angle such that when its vertex is placed at the center of a circle, the intercepted arc is equal in length to the radius. This unit is thus a little less than one of the angles of an equilateral triangle; in fact it follows from the geometry of the circle, since the length of a semicircumference is πr, that 1 RADIAN (1) π radians = 180°, where T = 3.14159, whence 1 radian = 57° 17' 44".806, or 57°.3 approximately. Inversely 1°.01745 radians. It is easy to change from degrees to radians and vice versa by means of relation (1), which should be remembered. Conversion tables for this purpose are printed in Table IV. FIG. 51 44. Use of Radian Measure. It is shown in geometry that two angles at the center of a circle are to each other as their intercepted arcs; therefore if an angle at the center is measured in radians and if the radius and the intercepted arc are measured in terms of the same linear unit, their numerical measures satisfy the simple relation : arc= angle × radius. * Sometimes also called circular measure. In other words, the number of linear units in the arc is equal to the product of the number of radians in the angle by the number of linear units in the radius. Example 1. Find the difference in latitude of two places on the same meridian 200 mi. apart, taking the radius of the earth as 4000 mi. 2° 51' 53", approximately. Angle = arc/radius = 1/20 in radians = 45. Angular Speed. In a rotating body a point P, which is at a distance from the axis of rotation, moves through a distance 2r during each revolution or through a distance r while the body turns through an angle of one radian. Therefore if is the linear (actual) speed of P (in linear units per time unit, e.g. feet per second), and if w is the angular speed of the rotating body (in radians per time unit, e.g. radians per second), then their numerical measures satisfy the relation v = r • w; hence the angular speed of a rotating body is numerically equal to the actual speed of a point one unit from the axis of rotation. Engineers usually express the angular speed of the rotating parts of machinery in revolutions per minute (R. P. M.) or revolutions per second (R. P. S.). These are easily reduced to radians per minute (or per second) by remembering that one revolution equals 2 π radians. Example 1. A flywheel of radius 2 ft. rotates at an angular speed of 2.5 R. P. S. Find the linear speed of a point on the rim. In radians per second, w = 2.5 × 2 π = 5π, and for a point 2 ft. from the axis of rotation v = 2 × 5 = 31.416 ft. per second. Example 2. Find the angular speed of a 34-in. wheel on an automobile going 20 mi. per hour. Every time the wheel turns through a radian the car goes forward 17 in. (the length of the radius), and 20 mi. per hour = 352 in. per sec.; therefore the wheel turns through 352/17 = 20.7 . . . radians per second. 46. Notation. In measuring angles in radian measure we shall adopt the practice universal in advanced work and write only the numerical measure of the angle in terms of the unit one radian. Thus in the expression tan x, the letter x will denote a number (the numerical measure of an angle), rather than the angle itself. When necessary, to call attention to the fact that radian measure is intended, the symbol () is appended to the numerical measure, thus: and so forth. 1)=1 radian = 57° 17' 44''.8, 2(r) = 2 radians = 114° 35' 29".6, As it happens that the acute angles whose trigonometric functions are most easily recalled without consulting tables are simple fractional parts of a straight angle, the number often appears as a factor of the numerical measure of angles. In this system, for example, sin (π/2) = = 1, cos (π/3) = 1/2, tan (π/4) = 1, etc. The use of pure numbers, such as 2 or in place of an angle, is precisely similar to the use of 10 for 10 ft. or 10 inches in expressing lengths. The student should supply the unit of measurement (radians, or feet or inches), and should not confuse the number π (= 3.14159...) with the angle whose measure is π radians, as he should not confuse the number 10 with the distance 10 feet. 47. Relations between Angular Units. The units of angle mentioned thus far are degree, minute, second, right angle, straight angle, revolution (or perigon), radian. The relations between these units is given in the following table: Another unit frequently used in France and occasionally elsewhere is the grade, which is 1/100 of a right angle. |