Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1905 edition. Excerpt: ...to use the triangle, as described in Fig. 63, its angles 90, 60, and 30, and its sides in the proportion of 2,1, and J3. To ascertain the numerical values of the sine, cosine, and tangent of 45, a similar method may be used. Thus, if AB and BC (Fig. 64) form two sides of a right-angled triangle in which BA=BC and each is one unit in length, the angle BAG--BCA, and as the sum of the two angles is 90 each angle is 45. Length of AC=s'AB3+BCi= Hence the three sides of the triangle ABC are in the ratio of 1, 1, and s/2;. AR0 BC 1 AB 1 sin45 =--=-=.: cos45 = =-t= AG-Jl AC VI or sin 45 = cos 45; tan 45 =-r7=L AB Angles greater than 90.--On p. 33 we have found that an angle is expressed by the amount of turning of a line such as AB (Fig. 65). If the movable radius, or line, occupies the positions AC, AG', AD, and AE, then it is seen that as ffC' = BC and the remaining sides of one triangle are equal to the corresponding sides of the other that the triangle BAG is equal to BAG'. Hence angle B'AC=(18Q"-30) = lo0, or sin 150=sin 30; or, generally, sin (180-l) = sin A. If the line AB be assumed to rotate in a negative direction until it reaches a point E, a negative angle equal to-30 is described; thus, the angle BAE may be written either as 330 or-30. In addition to the convention that all angles are measured in an anti-clockwise manner, the following rules are adopted: All lines measured in an upward direction from BB' are positive, and all lines measured from A'A towards B are positive; those in the opposite directions, i.e. downwards, or from A'A to B are negative. The movable radius, or line AC, or AC, is always positive. Hence, if BA C denote any angle A, then, since BC and EC are both in an upward direction, we have, as before sin (180-J)...