If-cos-A-B-1-2-sinA-1-2-then-what-is-cos-A-B-

Question Number 135926 by liberty last updated on 17/Mar/21

$$ \\ $$If cos(A+B) =1/2, sinA=1/√2, then what is cos (A – B)?

Answered by benjo_mathlover last updated on 17/Mar/21

$$\left(\mathrm{1}\right)\:\mathrm{cos}\:\left({A}+{B}\right)=\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{cos}\:\frac{\pi}{\mathrm{3}} \\ $$$$\Rightarrow{A}+{B}=\:\frac{\pi}{\mathrm{3}}+\mathrm{2}{n}\pi \\ $$$$\Rightarrow{B}\:=\:\frac{\pi}{\mathrm{3}}−{A}+\mathrm{2}{n}\pi \\ $$$$\Rightarrow\mathrm{cos}\:\left({A}−{B}\right)=\mathrm{cos}\:\left({A}−\frac{\pi}{\mathrm{3}}+{A}−\mathrm{2}{n}\pi\right) \\ $$$$\mathrm{cos}\:\left(\mathrm{2}{n}\pi+\frac{\pi}{\mathrm{3}}−\mathrm{2}{A}\right)=\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{3}}−\mathrm{2}{A}\right) \\ $$$$=\:\mathrm{cos}\:\frac{\pi}{\mathrm{3}}.\mathrm{cos}\:\mathrm{2}{A}+\mathrm{sin}\:\frac{\pi}{\mathrm{3}}.\mathrm{sin}\:\mathrm{2}{A} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} {A}\right)+\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}.\sqrt{\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{2}{A}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\mathrm{2}.\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\sqrt{\mathrm{1}−\left(\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} {A}\right)^{\mathrm{2}} } \\ $$$$=\:\mathrm{0}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}.\sqrt{\mathrm{1}−\left(\mathrm{1}−\mathrm{2}.\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$$=\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$

Commented by mr W last updated on 17/Mar/21

$$−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:{is}\:{also}\:{possible}. \\ $$

Answered by mr W last updated on 17/Mar/21

$$\mathrm{sin}\:\left({A}+{B}\right)=\pm\sqrt{\mathrm{1}−\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }=\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\mathrm{cos}\:{A}=\pm\sqrt{\mathrm{1}−\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} }=\pm\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$$$\mathrm{cos}\:\left({A}−{B}\right)=\mathrm{cos}\:\left[\mathrm{2}{A}−\left({A}+{B}\right)\right] \\ $$$$=\mathrm{cos}\:\mathrm{2}{A}\:\mathrm{cos}\:\left({A}+{B}\right)+\mathrm{sin}\:\mathrm{2}{A}\:\mathrm{sin}\:\left({A}+{B}\right) \\ $$$$=\left(\mathrm{1}−\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} {A}\right)\mathrm{cos}\:\left({A}+{B}\right)+\mathrm{2}\:\mathrm{sin}\:{A}\:\mathrm{cos}\:{A}\:\mathrm{sin}\:\left({A}+{B}\right) \\ $$$$=\left(\mathrm{1}−\mathrm{2}×\frac{\mathrm{1}}{\mathrm{2}}\right)×\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{2}×\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}×\left(\pm\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)×\left(\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right) \\ $$$$=\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$

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