Matematica: Esercizi di Trigonometria
$$sen\underbrace{\left(\displaystyle\frac{1}{2}x-\displaystyle\frac{\pi}{2}\right)}_{\alpha}=sen\underbrace{\left(\displaystyle\frac{1}{4}x-\displaystyle\frac{\pi}{4}\right)}_{\beta}$$Ricordando la definizione della funzione seno e della sua periodicità si ha:$$sen\alpha=sen\beta \Rightarrow$$$$ \alpha=\beta +2k\pi \lor \alpha + \beta = \pi + 2k\pi$$Per \(\alpha=\beta +2k\pi\) si ha: $$\displaystyle\frac{1}{2}x-\displaystyle\frac{\pi}{2}=\displaystyle\frac{1}{4}x-\displaystyle\frac{\pi}{4}+2k\pi \Rightarrow \displaystyle\frac{1}{4}x=\displaystyle\frac{\pi}{4}+2k\pi $$
$$x=\pi+8k\pi$$
Per \(\alpha+\beta= \pi +2k\pi\) si ha: $$\displaystyle\frac{1}{2}x-\displaystyle\frac{\pi}{2}+\displaystyle\frac{1}{4}x-\displaystyle\frac{\pi}{4}= \pi+2k\pi \Rightarrow $$$$ \displaystyle\frac{3}{4}x=\displaystyle\frac{3}{4}\pi+\pi+2k\pi$$
$$x=\displaystyle\frac{7}{3}\pi+\displaystyle\frac{8}{3}k\pi $$
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