# Find $\theta\in [0,2\pi]$ if $\theta = \frac{n\pi}{2}$ and $\theta = \frac{2m\pi}{5} + \frac{\pi}{10}$, where $n,m$ are integers

$$\theta = \dfrac{n\pi}{2}$$ and $$\theta = \dfrac{2m\pi}{5} + \dfrac{\pi}{10}$$ where $$n ,m \in \mathbb Z$$. Find $$\theta\in [0,2\pi]$$.

It can be solved by hit and try, of course but is there any faster way? Maybe concept of least common multiple can crack it? I tried that but I realised I was only taught to find LCM for natural numbers and not complicated equations. I found a problem which asked me to solve a trigonometric equation and I did until I got stuck here. Thanks a lot.

• Better to use words in your title instead of using Latex only. – StubbornAtom Apr 10 at 15:00
• @StubbornAtom can you please suggest a better title? Or give an example. – user541396 Apr 10 at 15:02
• As you can see I edited your title with some formatting adjustments. You can see this in the edit history. Related threads on Meta: math.meta.stackexchange.com/questions/9687/…, math.meta.stackexchange.com/questions/8891/…. – StubbornAtom Apr 10 at 15:05
• @StubbornAtom titles are supposed to be short so I usually don't add details like range of $\theta$ and unnecessary language. But I will take care of your suggestion next time. – user541396 Apr 10 at 15:11

$$\dfrac{n\pi}2=\dfrac{(4m+1)\pi}{10}$$

$$\iff5n=4m+1\iff5(n-1)=4(m+1)$$

$$\dfrac{4(m-1)}5=n-1$$ which is an integer

$$\implies5$$ divides $$4(m-1)$$

$$\implies 5$$ divides $$m-1$$ as $$(4,5)=1$$

WLOG $$m-1=5r$$ where $$r$$ is an integer

Can you take it from here?

• Can you please elaborate on the last step? – user541396 Apr 10 at 14:36
• Do you mean why $m-1=5r?$ – lab bhattacharjee Apr 10 at 14:38
• 4(m-1) should be a multiple of five , so. 4(m-1) = 5r where r is a integer..... I got that. But what happened next ? – user541396 Apr 10 at 14:45
• But $4,5$ don't have a common factor$>1$ – lab bhattacharjee Apr 10 at 14:46
• thank you got that . – user541396 Apr 10 at 14:51