Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T11:09:39.391Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.