Trig:How do deal with absolute value of trigonometric functions

[Tuharramah]

\(\displaystyle \left(x\, -\, \dfrac{\pi}{2}\right)^2\, \cdot\, \big|\cos(x)\, +\, \sin(x)\big|\, =\, \dfrac{\pi^2}{4}\, \big(\cos(x)\, +\, \sin(x)\big)\)

I went on expanding...

\(\displaystyle \left(x\, -\, \dfrac{\pi}{2}\right)^2\, =\, x^2\, -\, \pi\, +\, \dfrac{\pi^2}{4}\)

\(\displaystyle \left(x^2\, -\, \pi\, +\, \dfrac{\pi^2}{4}\right)^2\, \cdot\, \big|\cos(x)\, +\, \sin(x)\big|\, =\, \dfrac{\pi^2}{4}\, \big(\cos(x)\, +\, \sin(x)\big)\)

\(\displaystyle \left(x^2\, -\, \pi\, +\, \dfrac{\pi^2}{4}\right)^2\, \cdot\, \big|\cos(x)\, +\, \sin(x)\big|\, =\, \dfrac{\pi^2\, \cos(x)}{4}\, +\, \dfrac{\pi^2\, \sin(x)}{4}\)

Please give me tips

 

[ksdhart2]

\(\displaystyle \left(x\, -\, \dfrac{\pi}{2}\right)^2\, \cdot\, \big|\cos(x)\, +\, \sin(x)\big|\, =\, \dfrac{\pi^2}{4}\, \big(\cos(x)\, +\, \sin(x)\big)\)

I went on expanding...

\(\displaystyle \left(x\, -\, \dfrac{\pi}{2}\right)^2\, =\, x^2\, -\, \pi\, +\, \dfrac{\pi^2}{4}\)

\(\displaystyle \left(x^2\, -\, \pi\, +\, \dfrac{\pi^2}{4}\right)^2\, \cdot\, \big|\cos(x)\, +\, \sin(x)\big|\, =\, \dfrac{\pi^2}{4}\, \big(\cos(x)\, +\, \sin(x)\big)\)

\(\displaystyle \left(x^2\, -\, \pi\, +\, \dfrac{\pi^2}{4}\right)^2\, \cdot\, \big|\cos(x)\, +\, \sin(x)\big|\, =\, \dfrac{\pi^2\, \cos(x)}{4}\, +\, \dfrac{\pi^2\, \sin(x)}{4}\)

Please give me tips

What were the instructions given with this problem? For now I'll assume that it was to find any value(s) of x that make the equation true, for \(\displaystyle 0 \le x \le 2\pi\). In the future, please include the full and exact problem text. If necessarily, you can either translate it into English or leave it your original language and a helpful volunteer may be able to translate it.

One way you might deal with the absolute value is to break it apart into intervals. You know that \(\displaystyle |f(x)| = f(x) \text{ if } x \ge 0\) and \(\displaystyle |f(x)| = -f(x) \text{ if } x \le 0\). So, where is \(\displaystyle cos(x) + sin(x)\) positive? Where is it negative? When \(\displaystyle cos(x) + sin(x)\) is positive, what does that mean for your equation? What does it mean when it's negative? Does anything special happen when it's zero?