Question Number 73544 by Rio Michael last updated on 13/Nov/19
$${expressf}\left(\theta\right)=\:\mathrm{8}{cos}\theta\:−\mathrm{15}{sin}\theta\:{in}\:{the}\:{form} \\ $$ $$\:{rcos}\left(\theta\:+\:\alpha\right),\:{where}\:{r}>\mathrm{0}\:{and}\:\alpha\:{is}\:{a}\:{positive}\:{acute}\:{angle} \\ $$ $${hence} \\ $$ $${find}\:{the}\:{general}\:{solution}\:{of}\:{the}\:{equation} \\ $$ $$\:\:\mathrm{80}{cos}\:\theta\:−\mathrm{150}{sin}\theta\:=\:\mathrm{13} \\ $$ $${the}\:{maximum}\:{and}\:{minimum}\:{value}\:{of}\:\:\frac{\mathrm{5}}{{f}\left(\theta\right)\:+\:\mathrm{3}} \\ $$
Answered by mind is power last updated on 13/Nov/19
$$\mathrm{8}{cos}\left(\theta\right)−\mathrm{15}{sin}\left(\theta\right)=\mathrm{17}\left\{\frac{\mathrm{8}}{\mathrm{17}}{cos}\left(\theta\right)−\frac{\mathrm{15}}{\mathrm{17}}{sin}\left(\theta\right)\right\} \\ $$ $$=\mathrm{17}{cos}\left(\theta+{arcos}\left(\frac{\mathrm{8}}{\mathrm{17}}\right)\right) \\ $$ $$\alpha={arcos}\left(\frac{\mathrm{8}}{\mathrm{17}}\right) \\ $$ $$\mathrm{80}{cos}\left(\theta\right)−\mathrm{150}{sin}\left(\theta\right)=\mathrm{170}{cos}\left(\theta+\alpha\right)=\mathrm{13} \\ $$ $${cos}\left(\theta+\alpha\right)=\frac{\mathrm{13}}{\mathrm{170}} \\ $$ $$\theta+\alpha=\underset{−} {+}{arcos}\left(\frac{\mathrm{13}}{\mathrm{170}}\right)+\mathrm{2}{k}\pi \\ $$ $${min}\:{and}\:{max}\:\:{didnt}\:{exist} \\ $$ $${since}\:{f}\left(\theta\right)=−\mathrm{3}\:{hase}\:{solution} \\ $$
Commented byRio Michael last updated on 13/Nov/19
$${wonderful}\:{sir} \\ $$