A question on a error bound with trigonometric functions I have a link to a paper on a solution below
http://math.berkeley.edu/~zworski/128/psol07.pdf
This is related to my other question on the same problem.
For problem 7, the author achieves a second error bound by using
$$f^4(\xi) = | -119e^{2\xi}sin(3\xi) - 120e^{2\xi}cos(3\xi)|$$
and concludes
$$f^4(\xi) = | -119e^{2\xi}sin(3\xi) - 120e^{2\xi}cos(3\xi)| \le e^4\sqrt{119^2 + 120^2} \le 120\sqrt{2}e^4$$
What is the author doing here to get
$$e^4\sqrt{119^2 + 120^2}$$?
If someone could explain to me what is going on you'd save me a lot of time. I'm not sure what properties of trigonometric functions he's capitalizing on.
thanks!

Use that $a\sin x  + b\cos x = \sqrt{a^2 + b^2}\sin\left( x + \text{atan2}(b,a) \right)$.
