An Introduction to Galois Theory [Lecture notes] by Steven Dale Cutkosky

By Steven Dale Cutkosky

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1(a), (b), (c), (d), (e) and (f) respectively. cos x . 1 sin x The functions SIn x, cosec x = - . - , tan x = - - and cot x = - . , f(x) = - f( - x). e. f(x) = f( - x) . e. f(x) = f(x + 2nk) for all k E 71... e. f(x) = f(x + kn) for all k E Z. Sine waves The function f : x ..... a sin x + b cos x + c may be rewritten as f: x ..... R cos (x - a) + c. J(a 2 + b 2 ) , giving a unique value of a in the interval 0 ::::; a < 2n. 2 28 Curve sketching 2 (b) (a) cos x sin x x (d ) (c) cosec X Vi I I - 3,,/2 I I ,,/2 o 11\-1 I (e) ,,/2 sec x 2 1" I I I I !

By choosing k = 1 we are able to identify a unique value of a, called e, and for this function y = eX and :~ = e", The number e is irrational and its value is 2·71828 . . The function eX is called the exponential function and its graph is shown in Fig. 4(a). Transcendental curves 33 Working exercise On the same axes, sketch the graphs of y = 2\ y = eX and y = 3x • Note that all curves y = a" pass through the point (0, 1). The curve y = 2 X lies below that of y = eX for x > 0 and the curve y = 2X lies above y = eX for x < 0; the curve y = 3X lies above that of y = eX for x > 0 and below for x < O.

Are denoted by arcsin x, arctan x , etc. Working exercise Sketch the graphs of (a) y = SEC- 1 x and y = sec"! x, (b) y = COSEC- 1 x and y = cosec" x, (c) y = COT- 1 x and y = cot"! X. 3 Further parametric representations of curves Example 1 The ellipse b 2x 2 + a 2y2 = a 2b 2 (see Fig. 10(a), p. 19) is often expressed parametrically as x = a cos t , y = b sin t. Example 2 The hyperbola b 2x 2 - a 2y2 = a 2b 2 (see Fig. 1O(b), p. 19) is often expressed parametrically as x = a sec t, y = b tan t.

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