Question Number 125073 by amns last updated on 08/Dec/20
$${If}\:\sqrt{\mathrm{2}}{cos}\left({A}\:−\:{B}\right)\:=\:\mathrm{1},\:\mathrm{2}{sin}\left({A}\:+\:{B}\right)\:=\:\sqrt{\mathrm{3}}\:{and}\:{A},\:{B}\:{is}\:{a}\:{fine} \\ $$$${angle},\:{then}\:{determine}\:{A}'{s}\:{and}\:{B}'{s}\:{value}. \\ $$$$\boldsymbol{{Please}}\:\boldsymbol{{help}}\:\boldsymbol{{me}}. \\ $$
Answered by som(math1967) last updated on 08/Dec/20
$$\mathrm{cos}\left(\mathrm{A}−\mathrm{B}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}=\mathrm{cos45} \\ $$$$\left.\Rightarrow\mathrm{A}−\mathrm{B}=\mathrm{45}\:\:\:….\mathrm{i}\right) \\ $$$$\mathrm{sin}\left(\mathrm{A}+\mathrm{B}\right)=\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{sin30} \\ $$$$\left.\mathrm{A}+\mathrm{B}=\mathrm{30}\:\:\:…..\mathrm{ii}\right) \\ $$$$\left.\mathrm{i}\left.\right)+\mathrm{ii}\right) \\ $$$$\mathrm{2A}=\mathrm{75} \\ $$$$\mathrm{A}=\frac{\mathrm{75}}{\mathrm{2}} \\ $$$$\mathrm{B}=\mathrm{30}−\frac{\mathrm{75}}{\mathrm{2}}=−\frac{\mathrm{15}}{\mathrm{2}} \\ $$$$ \\ $$
Answered by mr W last updated on 08/Dec/20
![cos (A−B)=(1/( (√2))) ⇒A−B=2nπ±(π/4) sin (A+B)=((√3)/2) ⇒A+B=mπ+(−1)^m (π/3) ⇒A=(1/2)[2nπ±(π/4)+mπ+(−1)^m (π/3)] ⇒B=(1/2)[−2nπ∓(π/4)+mπ+(−1)^m (π/3)]](https://www.tinkutara.com/question/Q125082.png)
$$\mathrm{cos}\:\left({A}−{B}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$$$\Rightarrow{A}−{B}=\mathrm{2}{n}\pi\pm\frac{\pi}{\mathrm{4}} \\ $$$$\mathrm{sin}\:\left({A}+{B}\right)=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\Rightarrow{A}+{B}={m}\pi+\left(−\mathrm{1}\right)^{{m}} \frac{\pi}{\mathrm{3}} \\ $$$$\Rightarrow{A}=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{2}{n}\pi\pm\frac{\pi}{\mathrm{4}}+{m}\pi+\left(−\mathrm{1}\right)^{{m}} \frac{\pi}{\mathrm{3}}\right] \\ $$$$\Rightarrow{B}=\frac{\mathrm{1}}{\mathrm{2}}\left[−\mathrm{2}{n}\pi\mp\frac{\pi}{\mathrm{4}}+{m}\pi+\left(−\mathrm{1}\right)^{{m}} \frac{\pi}{\mathrm{3}}\right] \\ $$