By K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell (Auth.)
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Extra resources for Elementary Analysis. Volume 2
10. The domain is — 1 < Λ: < 1. In order to make sin-i χ single-valued we impose the restriction —90° < sin-i χ < 90°, and the graph is now restricted to the part of Fig. 10 which is shown by the thickened line. The range of = sin-^ χ will therefore be taken as -90°<>^<90°. Y4 FUNCTIONS OF COMPOUND ANGLES 37 Ex, 12. Suggest appropriate restrictions in range for the inverse functions cos"i X, tan-i x , and sketch the corresponding graphs. The circle x^+y^ Points on the circle -\- = = are represented parametrically by Λ: = fif cos y = a sin Θ.
13) 2 cos θ sin φ = sin (Θ + φ ) — sin (Θ — φ ) · (14) 2 cos θ cos φ = cos (Θ — φ ) + cos (Θ + φ)» (15) 2 sin θ sin φ = cos (Θ — φ ) — cos (Θ + φ ) · It is interesting to note that, before the invention of logarithms, these formulae were used to simplify the labour of multiplication. T h e m e t h o d is illustrated b y the following examples: 29-24 X 777-1 = 10^ χ 2 X 0-1462 X 0-7771, = 105 X 2 sin 8° 24'cos 39°, = 105 (sin 47° 2 4 ' - s i n 30° 36'), = 105(0-7361 - 0-5090), = 22710. T h e result using 4-figure logarithm tables is 22730.
Sin Β = 2 cos K A + B ) sin i(A - (18) cos A + cos Β = 2 cos K A + B ) cos K A (19) cos Β - B). cos A = 2 sin K A + B ) sin K A - Β)· This is a c u m b e r s o m e collection of formulae, but it is important to b e c o m e thoroughly familiar with them, as they are frequently used in m a n y branches of mathematics. T h e last set are often r e m e m b e r e d in words, for example, no. (19): cos minus cos is twice the sine of semi-sum times the sine of semi-difference reversed. 28 2 ELEMENTARY ANALYSIS VOLUME Although the authors are opposed to the memorization of a great number of formulae, and to excessive dependence on memory in the study of mathematics, it must be recognized that the student will find it impossible to think out suitable methods of developing many arguments that will arise in this subject without a reasonable famiharity with formulae (1) to (19) in the preceding pages.