 # I'm trying to find all intervals [ a , b ] on which the functions sin &#x20 anginih86 2022-06-24 Answered
I'm trying to find all intervals $\left[a,b\right]$ on which the functions $\mathrm{sin}\left(2\pi t\right)$ and $\mathrm{cos}\left(2pit\right)$ are orthogonal.
${\int }_{a}^{b}\mathrm{sin}\left(2\pi t\right)\cdot \mathrm{cos}\left(2\pi t\right)dt=\frac{\mathrm{cos}\left(4\pi b\right)-cos\left(4\pi a\right)}{8\pi }=0$
$\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\mathrm{cos}\left(4\pi b+2\pi k\right)=cos\left(4\pi a+2\pi l\right),k,l\in \mathbb{Z}$
I don't know how to solve this for a and b, can anybody help me with that please?
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$\frac{\mathrm{cos}\left(4\pi a\right)-\mathrm{cos}\left(4\pi b\right)}{8\pi }=0⟺$
$\mathrm{cos}\left(4\pi a\right)-\mathrm{cos}\left(4\pi b\right)=0⟺$
$-\mathrm{cos}\left(4\pi b\right)=-\mathrm{cos}\left(4\pi a\right)⟺$
$\mathrm{cos}\left(4\pi b\right)=\mathrm{cos}\left(4\pi a\right)⟺$

With ${n}_{1},{n}_{2}\in \mathbb{Z}$
So you can set:

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