Download e-book for iPad: Elementary Analysis. Volume 2 by K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell

By K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell (Auth.)

ISBN-10: 0080117775

ISBN-13: 9780080117775

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Additional resources for Elementary Analysis. Volume 2

Sample text

1 < sec JC < 2. -1 35. cosec ^x < 2. < tanjc < 1. 36. sin 2x < cos x. 37. Which of the following functions have the (i) same period, (ii) same amplitude? F o r each group state the common period or amplitude. (a) sin 2x, (b) 3 sin x, (c) COS(A: + 120°), (d) Kcos JC + V 3 sin x), (e) 3 cos 2JC, (f) sin(jc + 15°), (g) 3 COS(2JC + 4 5 ° ) , (h) i(cos JC + sin jc). 38. Sketch the graphs of the following functions within the domain 0 ° < Ö < 360°. (i) cos JC — sin JC, (ii) sin χ + sin 2x, (iii) cos X — cos ÍJC, (iv) 2 cos JC — 3 sin JC, (v) cos 2x — cos ix, (vi) 3 sin Jjc — 2 cos 2JC.

I) Show that, when χ is measured in radians, Ό χ sin αχ = a cos αχ, Όχ cos αχ = —a sin αχ. Write down the indefinite integrals of sin ax and cos ax. (ii) Show that D x tan χ = soc 2jc, using tan χ = sin x/cos JC. 3 shows a quadrant of a circle with centre O and unit radius. The area of the quadrant is ydx. 46 ELEMENTARY ANALYSIS VOLUME 2 where y = - x^. If ζ is the shaded area OMPB, and OM = x, then Όχζ = Put X = sin ^, so that y = cos and ΌβΖ = ΌχΖ . ΌβΧ = cos ^ . cos ^ = K l + cos 2^). Hence the area of the quadrant AO Β is (1 + cos 2Θ) άθ.

E X A M P L E . If y = 5 + 3 cos 9,find the greatest and least values of y and the values of θ for which they occur. The greatest value of cos Ö is + 1 , when θ = 360Λ°, and the least value of cos Ö is —1, when θ = (2n + 1) 180°. Hence the greatest value of 3; = 5 + 3 cos ^, is 8, when θ = 360/2°, and the least value is 2, when θ = (2n+ 1) 180°. In each case η represents an integer, positive, negative or zero. 3 1. If tan Ö = } and 180° < Ö < 270°, find sec θ and sin Θ. 2. If cosec θ = alb and 90° < Ö < 180°, find cot θ and cos B.

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Elementary Analysis. Volume 2 by K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell (Auth.)

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