# Generalized Trigonometric Intervals

1. May 25, 2015

### Cosmophile

1. The problem statement, all variables and given/known data

Identify the intervals of increase/decrease of $f(x) = \sin x + \cos x$

2. Relevant equations

$f(x) = \sin x + \cos x$

$f'(x) = \cos x - \sin x = \sqrt 2 \cos(x+\frac {\pi}{4})$

3. The attempt at a solution

$f$ is increasing when $f'(x) > 0$

$\sqrt 2 \cos(x+\frac {\pi}{4}) > 0 \to \cos(x + \frac {\pi}{4}) > 0$

So, $f$ is increasing on the intervals $(0, \frac {\pi}{4})$ and $(\frac {5 \pi}{4}, 2\pi)$, and is decreasing on the interval $(\frac {\pi}{4}, \frac {5 \pi}{4})$.

I know that in order to generalize this, you would add $2 \pi n$ to all intervals. I simply would like to know how this would be mathematically written. Thanks!

2. May 25, 2015

### micromass

Staff Emeritus
Like this: $f$ is decreasing on all intervals of the form $(\frac{\pi}{4} + 2\pi n, \frac{5\pi}{4} + 2\pi n)$ for each $n\in \mathbb{Z}$.

A more compact notation is available from set theory as

$$\bigcup_{n\in \mathbb{Z}} (\frac{\pi}{4} + 2\pi n, \frac{5\pi}{4} + 2\pi n)$$