Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-06T04:53:20.620Z Has data issue: false hasContentIssue false

Some Notes on Certain Theorems in Higher Trigonometry

Published online by Cambridge University Press:  03 November 2016

Extract

I Admire the ingenuity of the method by which Prof. Nanson establishes the expansions of sin x and cos x as power series in x. But I must confess that his attempt to ‘simplify’ the ‘accepted accurate’ proofs of these expansions seems to me a fundamentally mistaken one. Nor can I admit that the result is at all satisfactory ‘from the elementary didactic point of view.’ His proof is to my mind essentially an artificial verification, and altogether unnatural, in the sense that it has no place in any natural and logical way of developing the ideas which lead to the result. I speak with diffidence as one who has had far less experience of teaching than Prof. Nanson. But I am convinced that my criticism is a fair one: and I think that many of the proofs usually given, in English text-books, of many of the theorems of ‘Higher Trigonometry,’ are open to fair criticism on similar grounds, when (as is too seldom the case) they are not to be condemned for ‘hopeless complexity’ or utter lack of force.

Type
Research Article
Copyright
Copyright © Mathematical Association 1906 

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

* A modification of this method has, I find, been suggested by Prof. Bromwich (Math. Gaz., vol. iii., p. 85). There is a good deal to be said for the line of proof which he adopts, but I cannot.regard it as the best.

It has to be assumed that the exponential series can be differentiated term by term. I should never scruple, at this stage, to make assumptions such as this.