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The Power Series and the Infinite Products for sin x and cos x

Published online by Cambridge University Press:  03 November 2016

H. S. Carslaw
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Extract

1. The course of Plane Trigonometry “up to and including the solution of triangles and the properties of the circles associated with a triangle” offers little difficulty to the teacher and no serious difficulty to the pupil. But in the further development of the subject, when the circular functions sin x, cos x, … are regarded as functions of the real variable x, the position is different.

Type
Research Article
Information
The Mathematical Gazette , Volume 15 , Issue 206 , March 1930 , pp. 71 - 77
Copyright
Copyright © Mathematical Association 1930

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References

page 71 note * Cf. Bromwich, Infinite Series (2nd ed.), § 59.

page 72 note * If un >a and lim un exists, then lim un ≧a.

page 72 note † This theorem was given by J. Tannery in his Introduction à la Théorie des Fonctions d’uns Variable (2e éd, Paris, 1904), p. 292. See Bromwich, loc. cit., § 49.

page 74 note * This can be proved easily by the Differential Calculus; and a proof without the Calculus is to be found in Hobson’s Plane Trigonometry (7th ed.), p. 128.

page 74 note † We know that

Therefore

Now let

Then

Therefore

Thus

page 76 note * The absolute value of the expression under the logarithm has been taken to avoid having to work with logarithms of negative numbers.

Or we may take 0 < x < π in the first instance, and then extend the result.

page 76 note † The above footnote applies also here.

page 77 note * If we know that exists and is equal to α.