The linear combination of a sine and a cosine is itself a sine wave

A linear combination of two functions, f(x) and g(x) is a sum involving constant multiples of the functions.  That is,

a f(x)+b g(x)

where a, b \in \mathbb{R}.

So, in the case of \sin x and \cos x, we would have,

a\sin(x)+b\cos(x).

It is a slightly surprising fact that the linear combination of two sine waves is itself a sine wave.  The set of sine waves is closed under linear combination.

The in the featured animation, a\sin(x) is green and dotted and b\cos(x) is red and dotted.  The resulting linear combination is the continuous blue line.  The value a is set to 2.5, whereas the value b is animated.

This principle occurs in A level maths, Core 3, and is responsible for many long and complex questions.

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