Question Number 1812 by 112358 last updated on 04/Oct/15
$${Find}\:{the}\:{period}\:{of}\:{f}\left({x}\right)\:{where} \\ $$$${f}\left({x}\right)={cos}\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{3}}\right)+{sin}\left(\frac{\mathrm{3}{x}}{\mathrm{2}}−\frac{\pi}{\mathrm{4}}\right). \\ $$$${How}\:{may}\:{one}\:{find}\:{the}\:{period} \\ $$$${of}\:{the}\:{following}\:{functions} \\ $$$${if}\:\psi\left({x}\right)\:{has}\:{period}\:{m}\:{and}\:\gamma\left({x}\right) \\ $$$${has}\:{period}\:{n}? \\ $$$$\left(\mathrm{1}\right)\:{f}_{\mathrm{1}} \left({x}\right)=\psi\left({x}\right)\gamma\left({x}\right) \\ $$$$\left(\mathrm{2}\right)\:{f}_{\mathrm{2}} \left({x}\right)=\frac{\psi\left({x}\right)}{\gamma\left({x}\right)}\:\:\:\:\:\left\{\gamma\left({x}\right)\neq\mathrm{0}\right\} \\ $$$$\left(\mathrm{3}\right)\:{f}_{\mathrm{3}} \left({x}\right)=\left\{\psi\left({x}\right)\right\}^{\gamma\left({x}\right)} \\ $$$$\left(\mathrm{4}\right)\:{f}_{\mathrm{4}} \left({x}\right)={log}_{\psi\left({x}\right)} \gamma\left({x}\right) \\ $$$$\left(\mathrm{5}\right)\:{f}_{\mathrm{5}} \left({x}\right)=\psi\left({x}\right)\pm\gamma\left({x}\right) \\ $$$$\left(\mathrm{6}\right)\:{f}_{\mathrm{6}} \left({x}\right)={cos}\left(\psi\left({x}\right)\gamma\left({x}\right)\right) \\ $$$$ \\ $$
Answered by 123456 last updated on 06/Oct/15
$$\exists\left({s},{t}\right)\in\mathbb{Z}^{\mathrm{2}} ,{sm}={tn} \\ $$$$\left(\mathrm{1}\right),\left(\mathrm{2}\right)\:{and}\:\left(\mathrm{5}\right)\:{would}\:{have}\:{period}\:{lcm}\left({m},{n}\right) \\ $$$$ \\ $$