The calculus for engineers and physicists : integration and differentiation, with applications to technical problems and classified reference tables of integrals and methods of integration . ^ /R---N> s Fig. 14. 40 THE CALCULUS FOR ENGINEEKS. 78. Angle-Grradients of Tangent and Co-tangent and Integra-tion of Squares of Sine and Cosine.—In fig. 15, a small angle-increment 8a is marked off equally below and above the angle o,and radii are drawn from centre 0 through the extremities of 8aout to meet the two tangents to the quadrant of the circle at Nand E. The tangent of o, or tan a, is measur
The calculus for engineers and physicists : integration and differentiation, with applications to technical problems and classified reference tables of integrals and methods of integration . ^ /R---N> s Fig. 14. 40 THE CALCULUS FOR ENGINEEKS. 78. Angle-Grradients of Tangent and Co-tangent and Integra-tion of Squares of Sine and Cosine.—In fig. 15, a small angle-increment 8a is marked off equally below and above the angle o,and radii are drawn from centre 0 through the extremities of 8aout to meet the two tangents to the quadrant of the circle at Nand E. The tangent of o, or tan a, is measured along the tangent tan,(cf4Sa) >i* -itana— «i. from N to the radius at a, and its co-tangent, or cot a, along thetangent at E to the same radius. The increments of tan a, and ofcot a, due to 8a, are marked on the figure. 8 tan u, is a positiveincrease of tan o for a positive increase of the angle, while 8 cot ais a decrease of cot a. The lines U and ce are drawn parallel tothe short arc 8a. tt is therefore inclined to 8 tan a at the angle a,and cc to 8 cot o at the angle (90° — a). Therefore tt = cos tano and ce= - sin cot a. Here the - sign is used in order to make co positive (8 cot a beingnegative). Now tt is greater than 8a in the ratio of Ot to the radius of the circle, or =-^. Similarly ee is greater than 8a in the ratio -=-^.That is, 8a = ^^.cos a = (90° -a.) = ee sin , 80 = cos ^ tan a = - sin ^ cot a. EXAMPLES OF INTEGKATION AND DIFFERENTIATION. 41 Taking all the increments minutely small, these results are written (2 tan a —->- - =the angle-gradient of the tangent ]_ cos%and —,
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