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Question Number 123708 by mnjuly1970 last updated on 27/Nov/20

Commented by mnjuly1970 last updated on 27/Nov/20

please prove ⇑⇑

$${please}\:{prove}\:\Uparrow\Uparrow \\ $$

Answered by ajfour last updated on 27/Nov/20

((atan θ)/(cos θ))=h  (b/(sin θtan θ))=h  (a/(cos θ))+(b/(sin θ))=c  ⇒  asin θ=hcos^2 θ         bcos θ=hsin^2 θ  ⇒  asin θ+bcos θ=csin θcos θ=h  &   ab=h^2 sin θcos θ = (h^3 /c)  ⇒  abc=h^3

$$\frac{{a}\mathrm{tan}\:\theta}{\mathrm{cos}\:\theta}={h} \\ $$$$\frac{{b}}{\mathrm{sin}\:\theta\mathrm{tan}\:\theta}={h} \\ $$$$\frac{{a}}{\mathrm{cos}\:\theta}+\frac{{b}}{\mathrm{sin}\:\theta}={c} \\ $$$$\Rightarrow\:\:{a}\mathrm{sin}\:\theta={h}\mathrm{cos}\:^{\mathrm{2}} \theta \\ $$$$\:\:\:\:\:\:\:{b}\mathrm{cos}\:\theta={h}\mathrm{sin}\:^{\mathrm{2}} \theta \\ $$$$\Rightarrow\:\:{a}\mathrm{sin}\:\theta+{b}\mathrm{cos}\:\theta={c}\mathrm{sin}\:\theta\mathrm{cos}\:\theta={h} \\ $$$$\&\:\:\:{ab}={h}^{\mathrm{2}} \mathrm{sin}\:\theta\mathrm{cos}\:\theta\:=\:\frac{{h}^{\mathrm{3}} }{{c}} \\ $$$$\Rightarrow\:\:{abc}={h}^{\mathrm{3}} \\ $$

Commented by mnjuly1970 last updated on 27/Nov/20

bravo  bravo  master  ajfor.  thank you

$${bravo}\:\:{bravo}\:\:{master}\:\:{ajfor}. \\ $$$${thank}\:{you}\: \\ $$$$ \\ $$

Answered by ajfour last updated on 27/Nov/20

h^2 =((a/(cos θ)))((b/(sin θ)))   ...(i)  h(((sin θ)/(cos θ))+((cos θ)/(sin θ)))=c      ⇒  sin θcos θ=(h/c)     ...(ii)  (i)×(ii)  ⇒  h^3 =abc .

$${h}^{\mathrm{2}} =\left(\frac{{a}}{\mathrm{cos}\:\theta}\right)\left(\frac{{b}}{\mathrm{sin}\:\theta}\right)\:\:\:...\left({i}\right) \\ $$$${h}\left(\frac{\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta}+\frac{\mathrm{cos}\:\theta}{\mathrm{sin}\:\theta}\right)={c}\:\:\:\: \\ $$$$\Rightarrow\:\:\mathrm{sin}\:\theta\mathrm{cos}\:\theta=\frac{{h}}{{c}}\:\:\:\:\:...\left({ii}\right) \\ $$$$\left({i}\right)×\left({ii}\right)\:\:\Rightarrow\:\:{h}^{\mathrm{3}} ={abc}\:. \\ $$

Commented by mnjuly1970 last updated on 27/Nov/20

 god keep  you  young master    ′′ mr  ajfor ′′..

$$\:{god}\:{keep}\:\:{you}\:\:{young}\:{master} \\ $$$$\:\:''\:{mr}\:\:{ajfor}\:''.. \\ $$

Commented by ajfour last updated on 27/Nov/20

thanks, I wish the same for you.

$${thanks},\:{I}\:{wish}\:{the}\:{same}\:{for}\:{you}. \\ $$

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