Question Number 214351 by MATHEMATICSAM last updated on 06/Dec/24
Commented by MATHEMATICSAM last updated on 06/Dec/24
$$\mathrm{Find}\:{a} \\ $$
Commented by mr W last updated on 06/Dec/24
$${see}\:{also}\:{Q}\mathrm{214228} \\ $$
Commented by MATHEMATICSAM last updated on 07/Dec/24
$$\mathrm{Sir}\:\mathrm{I}\:\mathrm{understood}\:\mathrm{your}\:\mathrm{process}.\:\mathrm{can}\:\mathrm{you} \\ $$$$\mathrm{only}\:\mathrm{give}\:\mathrm{me}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{of}\:\mathrm{this}? \\ $$
Answered by mr W last updated on 07/Dec/24
Commented by mr W last updated on 07/Dec/24
$$\mathrm{cos}\:\theta=\frac{\mathrm{3}^{\mathrm{2}} +\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} }{\mathrm{2}×\mathrm{3}×\sqrt{\mathrm{2}}}=\frac{\mathrm{7}\sqrt{\mathrm{2}}}{\mathrm{12}} \\ $$$$\Rightarrow\mathrm{sin}\:\theta=\frac{\sqrt{\mathrm{46}}}{\mathrm{12}} \\ $$$${a}^{\mathrm{2}} =\mathrm{3}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} −\mathrm{2}×\mathrm{3}×\mathrm{1}\:\mathrm{cos}\:\left(\mathrm{45}°+\theta\right) \\ $$$$\:\:\:\:=\mathrm{10}−\mathrm{6}\:\left(\mathrm{cos}\:\mathrm{45}°\:\mathrm{cos}\:\theta−\mathrm{sin}\:\mathrm{45}°\:\mathrm{sin}\:\theta\right) \\ $$$$\:\:\:\:=\mathrm{10}−\frac{\mathrm{6}}{\:\sqrt{\mathrm{2}}}\:\left(\frac{\mathrm{7}\sqrt{\mathrm{2}}}{\mathrm{12}}−\frac{\sqrt{\mathrm{46}}}{\mathrm{12}}\right) \\ $$$$\:\:\:\:=\frac{\mathrm{13}+\sqrt{\mathrm{23}}}{\mathrm{2}} \\ $$$$\Rightarrow{a}=\frac{\sqrt{\mathrm{2}\left(\mathrm{13}+\sqrt{\mathrm{23}}\right)}}{\mathrm{2}}\approx\mathrm{2}.\mathrm{983} \\ $$
Commented by MATHEMATICSAM last updated on 07/Dec/24
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 07/Dec/24
$${for}\:{general}\:{formula}\:{see}\:{Q}\mathrm{130241} \\ $$