Question Number 119635 by Ar Brandon last updated on 25/Oct/20 $$\mathrm{If}\:\mathrm{a}\:\mathrm{function}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{relation} \\ $$$$\:\:\:\:\:\:\:{f}\left(\mathrm{x}+\mathrm{1}\right)+{f}\left(\mathrm{x}−\mathrm{1}\right)=\sqrt{\mathrm{3}}{f}\left(\mathrm{x}\right)\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}\in\mathbb{R} \\ $$$$\mathrm{then}\:\mathrm{a}\:\mathrm{period}\:\mathrm{of}\:{f}\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{10}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{12}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{4} \\ $$ Answered by Olaf last updated on…
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Question Number 54093 by Tawa1 last updated on 28/Jan/19 $$\mathrm{If}\:\:\:\:\:\:\mathrm{tan}\left(\mathrm{z}\right)\:\:=\:\:\mathrm{2},\:\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{z} \\ $$ Commented by mr W last updated on 28/Jan/19 $${z}=\mathrm{tan}^{−\mathrm{1}} \mathrm{2}+{n}\pi \\ $$ Commented…
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Question Number 185151 by mathlove last updated on 17/Jan/23 Answered by mr W last updated on 20/Jan/23 $${say}\:{the}\:{touching}\:{point}\:{with}\:{the} \\ $$$${parabola}\:{is}\:\left(−{p},\frac{{p}^{\mathrm{2}} }{\mathrm{2}}\right). \\ $$$${the}\:{center}\:{of}\:{circle}\:{is}\:\left(−{R},{h}\right). \\ $$$$\mathrm{tan}\:\theta={p}…
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Question Number 185137 by emmanuelson123 last updated on 17/Jan/23 Answered by CElcedricjunior last updated on 17/Jan/23 $$\boldsymbol{{montrons}}\:\boldsymbol{{que}}\:\forall\boldsymbol{{n}}\in\boldsymbol{{IN}} \\ $$$$\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\left[\left(\mathrm{7}+\mathrm{4}\sqrt{\mathrm{3}}\right)^{\boldsymbol{{n}}} −\left(\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}\right)^{\boldsymbol{{n}}} \right]\equiv\mathrm{0}\left[\mathrm{6}\right] \\ $$$$\boldsymbol{{posons}}\:\boldsymbol{{p}}\left(\boldsymbol{{n}}\right)=\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\left[\left(\mathrm{7}+\mathrm{4}\sqrt{\mathrm{3}}\right)^{\boldsymbol{{n}}} −\left(\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}\right)^{\boldsymbol{{n}}} \right]…
Question Number 185128 by Shrinava last updated on 17/Jan/23 Answered by Frix last updated on 17/Jan/23 $$\mathrm{Formulas}\:\mathrm{for}\:\mathrm{incircle}\:\mathrm{and}\:\mathrm{3}\:\mathrm{excircles}: \\ $$$${r}=\frac{\mathrm{2}{A}}{{a}+{b}+{c}} \\ $$$${r}_{{a}} =\frac{\mathrm{2}{A}}{{b}+{c}−{a}} \\ $$$${r}_{{b}} =\frac{\mathrm{2}{A}}{{a}+{c}−{b}}…