# 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.