Fie \( M \) o multime de numere reale cu proprietatile:
1) \( 0\in M \);
2) \( x\in M\Rightarrow \sin x+\cos x\in M \);
3) \( \sin \ 2x+\cos \ 2x\in M\Rightarrow x\in M \).
Sa se arate ca:
a) \( \frac{3\pi}{4}\in M \);
b) M contine o infinitate de numere irationale subunitare.
Lucian Dragomir, RMT 2002
O multime de numere reale
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O multime de numere reale
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Marius Mainea
- Gauss
- Posts: 1077
- Joined: Mon May 26, 2008 2:12 pm
- Location: Gaesti (Dambovita)
a) \( 1=\sin 0+\cos 0 \in M \)
\( \sin 2\pi+\cos 2\pi=1\in M \) deci \( \pi\in M \)
\( -1=\sin \pi+\cos \pi \in M \)
\( -1=\sin\frac{3\pi}{2}+\cos\frac{3\pi}{2}\in M \) deci \( \frac{3\pi}{4}\in M \)
b) se arata prin inductie ca daca \( x\in M \) atunci \( \frac{x}{2^n}\in M (\forall) n\in\mathbb{N} \)
\( \sin 2\pi+\cos 2\pi=1\in M \) deci \( \pi\in M \)
\( -1=\sin \pi+\cos \pi \in M \)
\( -1=\sin\frac{3\pi}{2}+\cos\frac{3\pi}{2}\in M \) deci \( \frac{3\pi}{4}\in M \)
b) se arata prin inductie ca daca \( x\in M \) atunci \( \frac{x}{2^n}\in M (\forall) n\in\mathbb{N} \)