נתון ( ∙ {\displaystyle \bullet } ): א. ∠ C = 90 ∘ B D = C D = X {\displaystyle {\begin{aligned}&\angle C=90^{\circ }\\&BD=CD=X\\\end{aligned}}} ב. A D = 10 c m ∠ β = 50 ∘ {\displaystyle {\begin{aligned}&AD=10_{cm}\\&\angle \beta =50^{\circ }\\\end{aligned}}} צ"ל : t a n β = 2 tan α {\displaystyle \ tan\beta =2\tan \alpha }
הוכחה : ∠ C = 90 ∘ tan α = A C B C ∙ B C = 2 x tan α = A C x A C = tan α ∗ 2 x tan β = A C D C ∙ D C = X tan β = A C x A C = tan β ∗ x tan α ∗ 2 x = tan β ∗ x / : X 2 tan α = tan β ◼ 2 tan 50 ∘ = tan β β = 67.239 ∙ A D = 10 c m cos β = D C A D cos 67.239 = X 10 ↓ x = 3.869 2 x = B C = 7.738 {\displaystyle {\begin{aligned}&\angle C=90^{\circ }\\&\tan \alpha ={\frac {AC}{BC}}\\&\bullet BC=2x\\&\tan \alpha ={\frac {AC}{x}}\\&\color {blue}AC=\tan \alpha *2x\\&\tan \beta ={\frac {AC}{DC}}\\&\bullet DC=X\\&\tan \beta ={\frac {AC}{x}}\\&\color {blue}AC=\tan \beta *x\\&\tan \alpha *2x=\tan \beta *x/:X\\&2\tan \alpha =\tan \beta \blacksquare \\&2\tan 50^{\circ }=\tan \beta \\&\beta =67.239\\&\bullet AD=10_{cm}\\&\cos \beta ={\frac {DC}{AD}}\\&\cos 67.239={\frac {X}{10}}\\&\downarrow \\&x=3.869\\&2x=BC=7.738\\\end{aligned}}}
