By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

ISBN-10: 9810223854

ISBN-13: 9789810223854

Those chosen papers of S.S. Chern talk about issues corresponding to quintessential geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional area, and transgression in linked bundles

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**Example text**

But the last two are roots of - (cos e + sin 0) = 1. ) 1 Transform the fol1owing into the form stated, giving the values of rand y. /3cos 0 - sinO, rcos(O + y); (c) cos 20 - sin 20, rcos(20 + y); (d) 3sin30-cos30, rsin(30 + y). 2 Sketch the graphs of the expressions in Question I. 3 Solve the fol1owing equations for values of 0 between 00 and 3600 by each of the method s described on pp . /2; (b) 2cosO - sinO = 2 ; (c) 7sinO - 24cosO = lO. 4 Find the maximum and minimum values of the fol1owing expressions.

Since IX is in the range [ -n/2, n/2] , it follows that IX = -n/2. Example 3 Prove that tan " (t) Let 30 Methods of Trigonometry + Ztan " m= coe 1 (37/39). y y = Tan- Ix x Fig . 4 tan -1 (t) ~ tan " =¢ ~ m+ 2 tan " tan ¢ (t) = t, = () + 2¢. ) tan (() + 2'I' 2¢'-= -tan- 0-+ -tan-----' 1 - tan 0 tan 2¢ and tan 2¢ 2 tan ¢ 2 x t 8 = 1 _ tan ? ¢ = 1 - 1/16 = 15 ~ tan (0 + 2¢) =;> 0 t + 8/ 15 39 = 1 _ t(8/15) = 37 + 2¢ = tan "! (39/37) = cot " (37/39). Note that here we have used the result tan " x = cot " (l/x) .

1 Approximations for small angles Consider a small angle 0 subtended by an arc AB at the centre 0 of a circle of radius r (see Fig. 1). Draw AC perpendicular to OA to cut OB at C. Then AC=rtanO and area 60AC=tr 2tan{). If we draw the chord AB then 60AB is an isosceles triangle of area tr 2 sin O. 2) on p. 2, it follows that tr 2 sin 0 < tr 2 () < tr 2 tan 0 = sin {) < () < tan {). = I < ---'---0 < sec O. 2) ---'---0 -+ 1. 1) by tan 0 = cos o < ~{) < tan 1. 3) ---+1. --l~ A Fig. 4) sinO ~ 0, tan 0 ~ O.