Non-linearity of Sine

While at a meeting yesterday the following question came up due to a student “Why isn’t \(\mathrm{sin}(a+b) = \mathrm{sin}(a) + \mathrm{sin}(b) \sin\)“. My immediate response would be to say “because \(\sin(x)\) isn’t a linear function”, but this isn’t a terribly satisfactory response since the likelihood is that the student doesn’t have a deep understanding of the difference between linear and non-linear functions.

On reflection I think the best explanation is to ask the student to sketch the graph of \(\sin(x)\) and ask them to consider \(\sin(a+b)\) using the graph. singraph

It should become clear from the graph that \(\mathrm{sin}(a+b) \) can only be equal to \(\mathrm{sin}(a) + \mathrm{sin}(b)\) when either one of \(a\) or \(b\) is \(360^\circ\) (in fact any multiplicity of the periodicity of \(\sin(x)\).

Another approach could be to ask them to consider a sequence of non-linear functions such as \(f(x)=x^2, f(x) = x^3+3,  f(x) = 2^x+6\) and ask them to compute \(f(a),f(b),f(a+b)\). This I hope would get rid of the expectation that f(a+b) = f(a)+f(b). The geometric proofs of the true formulae for \(\sin(a+b)\) could be a nice way to close the discussion.

I wonder if this is a consequence of an over-reliance on linear functions for examples of substitution etc in lower school.

Leave a Reply

Your email address will not be published. Required fields are marked *