Question Number 20653 by Tinkutara last updated on 30/Aug/17 $$\mathrm{Length}\:\mathrm{of}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{range}\:\mathrm{of}\:\mathrm{function} \\ $$$${f}\left(\theta\right)\:=\:\mathrm{cos}^{\mathrm{2}} \:\theta\:−\:\mathrm{6}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:+\:\mathrm{3}\:\mathrm{sin}^{\mathrm{2}} \:\theta\:+\:\mathrm{2} \\ $$$$\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{8} \\ $$$$\left(\mathrm{2}\right)\:−\mathrm{8} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\mathrm{10}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\sqrt{\mathrm{10}} \\…
Question Number 86189 by oustmuchiya@gmail.com last updated on 27/Mar/20 $${Given}\:{that}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\frac{\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{7}}{\mathrm{8}}\:{and}\:\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{x}}\right)=\:\frac{\mathrm{3}\boldsymbol{\mathrm{x}}−\mathrm{6}}{\mathrm{6}},\:{find} \\ $$$$\left(\mathrm{a}\right)\:\boldsymbol{\mathrm{g}}\left(\mathrm{6}\right),\:\left(\mathrm{b}\right)\boldsymbol{\mathrm{f}}^{−\mathrm{1}} \left(\boldsymbol{\mathrm{x}}\right)\:\:\:\left(\mathrm{c}\right)\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\boldsymbol{\mathrm{x}}\:{if}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{x}}\right) \\ $$ Answered by Kunal12588 last updated on 27/Mar/20 $$\left({a}\right)\:{g}\left(\mathrm{6}\right)=\frac{\mathrm{18}−\mathrm{6}}{\mathrm{6}}=\mathrm{2} \\ $$$$\left({b}\right)\:{f}^{−\mathrm{1}}…
Question Number 20652 by Tinkutara last updated on 30/Aug/17 $$\mathrm{Suppose}\:{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left\{{x}\right\},\:\left[{x}\right]\:\mathrm{and}\:{x}\:\mathrm{are}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{geometric}\:\mathrm{progression}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{such}\:\mathrm{that}\:{x}^{{n}} \:>\:\mathrm{100}. \\ $$$$\left(\mathrm{Here}\:\left[{x}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{part}\:\mathrm{of}\:{x}\right. \\ $$$$\left.\mathrm{and}\:\left\{{x}\right\}\:=\:{x}\:−\:\left[{x}\right].\right) \\ $$ Commented by…
Question Number 151721 by mathdanisur last updated on 22/Aug/21 Commented by liberty last updated on 22/Aug/21 $$\:\mathrm{x}=\mathrm{y}=\mathrm{z}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 20648 by Joel577 last updated on 30/Aug/17 $$\mathrm{If}\:\:\frac{{F}}{\mathrm{sin}\:\left({A}\:−\:{a}\right)}\:=\:\frac{{W}}{\mathrm{sin}\:{A}},\:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{tan}\:{A}\:=\:\frac{{W}\:\mathrm{sin}\:{a}}{{W}\:\mathrm{cos}\:{a}\:−\:{F}} \\ $$ Answered by $@ty@m last updated on 30/Aug/17 $${Given}, \\ $$$$\:\frac{{F}}{\mathrm{sin}\:\left({A}\:−\:{a}\right)}\:=\:\frac{{W}}{\mathrm{sin}\:{A}} \\…
Question Number 20647 by oyshi last updated on 30/Aug/17 $${prove}\:{it}, \\ $$$$\mathrm{tan}\:\left(\alpha+\frac{\pi}{\mathrm{3}}\right)+\mathrm{tan}\:\left(\alpha−\frac{\pi}{\mathrm{3}}\right)=\frac{\mathrm{4sin}\:\mathrm{2}\alpha}{\mathrm{1}−\mathrm{4sin}\:^{\mathrm{2}} \alpha} \\ $$ Answered by $@ty@m last updated on 30/Aug/17 $${LHS}=\frac{{tan}\alpha+{tan}\frac{\pi}{\mathrm{3}}}{\mathrm{1}−{tan}\alpha.{tan}\frac{\pi}{\mathrm{3}}}+\frac{{tan}\alpha−{tan}\frac{\pi}{\mathrm{3}}}{\mathrm{1}+{tan}\alpha.{tan}\frac{\pi}{\mathrm{3}}} \\ $$$$=\frac{{tan}\alpha+\sqrt{\mathrm{3}}}{\mathrm{1}−\sqrt{\mathrm{3}}.{tan}\alpha}+\frac{{tan}\alpha−\sqrt{\mathrm{3}}}{\mathrm{1}+\sqrt{\mathrm{3}}.{tan}\alpha}…
Question Number 20646 by oyshi last updated on 30/Aug/17 $${if}\:\mathrm{cos}\:\left(\beta−\gamma\right)+\mathrm{cos}\left(\:\gamma−\alpha\right)+\mathrm{cos}\:\left(\alpha−\beta\right)=−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${so}\:{proof}\:{it}, \\ $$$$\Sigma\mathrm{cos}\:\alpha=\mathrm{0},\Sigma\mathrm{sin}\:\alpha=\mathrm{0} \\ $$ Answered by $@ty@m last updated on 31/Aug/17 $${Given}, \\…
Question Number 151714 by iloveisrael last updated on 22/Aug/21 Commented by liberty last updated on 22/Aug/21 $$\:\mathrm{d}=\frac{\mathrm{8}}{\:\sqrt{\mathrm{5}}}\:\mathrm{cm} \\ $$ Commented by liberty last updated on…
Question Number 20640 by oyshi last updated on 30/Aug/17 $$\mathrm{sin}\:\theta={K}\mathrm{cos}\:\left(\theta−\alpha\right) \\ $$$${so}\:{proof}\:{it},\mathrm{cot}\:\theta=\frac{\mathrm{1}−{K}\mathrm{sin}\:\alpha}{{K}\mathrm{cos}\:\alpha} \\ $$ Answered by mrW1 last updated on 30/Aug/17 $$\mathrm{sin}\:\theta=\mathrm{K}\left(\mathrm{cos}\:\theta\:\mathrm{cos}\:\alpha+\mathrm{sin}\:\theta\:\mathrm{sin}\:\alpha\right) \\ $$$$\left(\mathrm{1}−\mathrm{Ksin}\:\alpha\right)\mathrm{sin}\:\theta=\mathrm{Kcos}\:\alpha\:\mathrm{cos}\:\theta \\…
Question Number 20639 by oyshi last updated on 30/Aug/17 $${if}\:\mathrm{cos}\:\left({A}+{B}\right)\mathrm{sin}\:\left({C}+{D}\right)=\mathrm{cos}\:\left({A}−{B}\right) \\ $$$$\mathrm{sin}\:\left({C}−{D}\right) \\ $$$${so}\:{proof}\:\mathrm{cot}\:{A}\mathrm{cot}\:{B}\mathrm{cot}\:{C}=\mathrm{cot}\:{D} \\ $$ Answered by Tinkutara last updated on 30/Aug/17 $$\frac{\mathrm{sin}\:\left({C}+{D}\right)}{\mathrm{sin}\:\left({C}−{D}\right)}=\frac{\mathrm{cos}\:\left({A}−{B}\right)}{\mathrm{cos}\:\left({A}+{B}\right)} \\…