Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . nown for rectifying a semi-circumference. Accord-ing to Bottcher it is due to a Polish Jesuit, Kochansky, and was published in the Acta Eruditorum Llpsiae, 16So. The demon-stration is as follows ; Calling the radius unity, the diameter would have the numerical value 2. Then in Fig. 2, Plate I, we have on = y/oh- -f lin- = \/ok- + {kn — k/i)- = \/4 -)- (3 — tan 30°)2 = -fThe tangent of an angle (abbreviated to tan.) is a trigonometric fun
Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . nown for rectifying a semi-circumference. Accord-ing to Bottcher it is due to a Polish Jesuit, Kochansky, and was published in the Acta Eruditorum Llpsiae, 16So. The demon-stration is as follows ; Calling the radius unity, the diameter would have the numerical value 2. Then in Fig. 2, Plate I, we have on = y/oh- -f lin- = \/ok- + {kn — k/i)- = \/4 -)- (3 — tan 30°)2 = -fThe tangent of an angle (abbreviated to tan.) is a trigonometric function whose numerical value can be obtained froma table. A draughtsman has such freguent occasion to use these functions that they are given here for reference, both as lines,and as ratios. Trigonometric Functions as Eatios. Trigonometric Functions as Lines. e = the given angle = CA B h = hypothenuse of triangle CAB a = A B = side of triangle adjacent to vertex of 9 o = £C=side of triangle opposite to e Then sin . 9 = j-; cos i — tt ; Cu-tangent of Q tan e = cotan Q - sm I cos I : reciprocal of cosine. cos I bin ( : reciprocal of tan A«^ B The prefix co suggests complement; the co-sine of B is the sine of the complement of 5, &c. As lines the functions-may be defined as follows : The sine of an arc (e. g., that subtended by angle 6 in the figure) is the perpendicular {CB) let fall from one extremity ofthe arc upon the diameter passing through the other extremity. If the radius A C, through one extremity of the arc, beprolonged to cut a line tangent at the other extremity, the intercepted portion of the tangent is called the tangent of the arc,and the distance, on such extended radius, from the centre of the circle to the tangent, is called the secant of the arc. The co-sine, co-secant and co-tangent of the arc are respectively the sine, secant and tangent of the complement of the-given arc. 32 THEORETICAL AND PRACTICAL GRAPHICS. a and be tangent to the line mv?^. D
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