If-and-are-the-interior-angles-of-a-triangle-find-the-value-of-determinant-tan-1-1-1-tan-1-1-1-tan-

Question Number 139484 by ZiYangLee last updated on 27/Apr/21

If α,β and γ are the interior angles of a triangle, find the value of determinant (((tan α),1,1),(1,(tan β),1),(1,1,(tan γ)))

$$\mathrm{If}\:\alpha,\beta\:\mathrm{and}\:\gamma\:\mathrm{are}\:\mathrm{the}\:\mathrm{interior}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{a}\: \\ $$$$\mathrm{triangle},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{vmatrix}{\mathrm{tan}\:\alpha}&{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{tan}\:\beta}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{1}}&{\mathrm{tan}\:\gamma}\end{vmatrix} \\ $$

Answered by MJS_new last updated on 27/Apr/21

2 all you need is tan γ =tan (π−(α+β)) =−tan (α+β) = =−((tan α +tan β)/(1−tan α tan β))

$$\mathrm{2} \\ $$$$\mathrm{all}\:\mathrm{you}\:\mathrm{need}\:\mathrm{is} \\ $$$$\mathrm{tan}\:\gamma\:=\mathrm{tan}\:\left(\pi−\left(\alpha+\beta\right)\right)\:=−\mathrm{tan}\:\left(\alpha+\beta\right)\:= \\ $$$$=−\frac{\mathrm{tan}\:\alpha\:+\mathrm{tan}\:\beta}{\mathrm{1}−\mathrm{tan}\:\alpha\:\mathrm{tan}\:\beta} \\ $$

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