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

A linear combination of two functions, and is a sum involving constant multiples of the functions.  That is,

where .

So, in the case of and , we would have,

.

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, is green and dotted and is red and dotted.  The resulting linear combination is the continuous blue line.  The value is set to 2.5, whereas the value is animated.

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