# sin(x) + cos(x)

Here is a fun little exploration involving a simple sum of trigonometric functions.

Consider *f(x) = *sin(*x*)* + *cos(*x*), graphed below.

Surprisingly, it appears as though sin(*x*)* + *cos(*x*) is itself a sine function. And while its period is the same as sin(*x*), its amplitude has changed and it’s been phase-shifted. Figuring out the exact amplitude and phase shift is fun, and it’s also part of a deeper phenomenon to explore.

Consider the function *g*(*x*) *= a* sin(*x*)* + b cos*(*x*). Playing around with the values of *a* and *b *is a great way to explore the situation.

On the way to a complete solution, a nice challenge is to find (and characterize) the values of *a* and *b* that make the amplitude of g*(x)* equal to one. It’s also fun to look for values of *a* and *b* that yield integer amplitudes: for example, 5sin(*x*)* + *12cos(*x*)* *has amplitude 13, and 4sin(*x*)* +* 3cos(*x*)* *has amplitude 5.

Ultimately, this exploration leads to a really lovely application of angle sum formulas. Recall that

If we let *A* = *x*, we get

With a little rewriting, we have

which looks similar to our original function *f(x) = *sin(*x*)* + *cos(*x*), except for what’s in front of sin(*x*)* *and cos(*x*). We handle that with a clever choice of *B*.

Let . Now we have

And a little algebra gets us

And so sin(x) + cos(x) really is a sine function! Not only does this transformation explain the amplitude and phase shift of sin(x) + cos(x), it generalizes beautifully.

For example, consider 5sin(*x*)* + *12cos(*x*). We can rewrite this in the following way.

where .

There’s quite a lot of trigonometric fun packed into this little sum. And there’s still more to do, like exploring different phase shifts and trying the cosine angle sum formula instead. Enjoy!

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