COR. 1. Since cor; is a constant quantity. r This constant quantity is always represented by 2π; the approximate value of π, (3.14159...,) will be determined hereafter. COR. 2. Since = 2π; .. c=2πr; or 2πr represents the circumference of a circle whose radius is r. 100. If an arc be traced out by a point in the line CB, by whose revolution from the position CA an angle ACB is described, the angle ACB may be properly measured by arc AB the ratio radius AC Since in equal circles, and therefore in B the same circle, angles at the centre have the same ratio to each other as the arcs on which they stand, Euclid vi. 33; or decreases in proportion as the angle ACB increases or decreases; therefore arc is a proper measure of the mag radius nitude of an angle, and we may say that the angle is equal 101. In the preceding Chapters we have measured the magnitude of an angle by the number of times it contains a fixed and definite angle called a degree, and its subdivisions;— for several analytical investigations, however, the circular is much more convenient. measure arc radius When this circular measure is used, we shall, generally speaking, denote angles by the letters of the Greek Alphabet. 102. Having given the circular measure arc radius of an angle, to determine how many degrees the angle contains; and conversely. Let be the circular measure of the angle which contains A degrees. circumference Since = = 2π, and the circumference subtends radius four right angles, therefore 2π is the circular measure of four right angles. Ex. 2. Required the number of degrees subtended by the are which is equal to the radius. And the degrees, minutes, and seconds required are 57°, 17′, 44"77. COR. It is often necessary to know the number of seconds subtended by an arc which is equal to the radius. The number of degrees subtended = 57.29577 minutes 3437.7462 60 60 number of seconds subtended=206264•772=206264-8 nearly. 103. Comparison of the unit of measurement in the two methods of estimating the magnitude of an angle. If the arc subtending the angle be equal in length to arc the radius with which it is described, the expression radius becomes 1, and therefore this is the angle which is the unit of measurement when the magnitudes of angles are estimated by the circular measure. Now this angle, as it appears from Ex. 2, of the last Article, contains the angle which is called a degree 57.29577 times; and therefore the one of these angular units is 57-29577 times greater than the other. 104. Four right angles being represented by 2π, and therefore two right angles by π, we have from Arts. 24, 25, if the angle A be represented according to the circular measure by 0, (cos +/- 1 sin 0)" = cos m 0 ±√√√-1 sin m0. (Cos 0-1 sin 0). (cos +/-1 sin 0) = (cos 0) — (sin ()2 ± √ − 1 . 2 sin 0 . cos 0. Or, (cos + v 1 sin 0) = cos 20 ± √ - 1 sin 26. M Again, (cos +√1 sin 0). (cos + √1 sin 0) = (cos 20√1 sin 20). (cos + sin 0) = = (cos20.cos-sin 20. sin 0) ±√-1. (sin20. cose+cos20. sin(); .. (cos±√1 sin 0)3 = cos 30 ± √1 sin 30. Suppose this law to hold for m factors, so that (cos + 1 sin ()" = cos me ± - 1 sin m0; (cos me±√1 sin me). (cos±√1 sin 0) v = cos me.cos - sin me. sin 0 ±√-1. (sinme.cos + cos m. sinė) = cos (m + 1) 0 ± √ - 1. sin (m + 1) 0. If therefore the law hold for m factors, it holds for m + 1 factors; but we have shewn it to hold when m = 3, it therefore holds when m= 4, and by successive inductions we conclude it to be true when the index is any positive integer. |