Question

$\text{Sa se rezolve ecuatia} \\ \\\\sin(x-\dfrac{\pi}{4}})=1 \\ \\\text{Trebuie rezolvata cu formula aceasta : }\boxed{(-1)^k~arcsin~a~+k\pi~~| ~k \in Z}$

Answer 1

Se utilizeaza formula trigonometrica cunoscuta

sin( x -π/4 ) = sin x ·cos( π/4 ) -sin( π/4 ) ·cos x <=>

sin( x -π/4 ) = sin x ·√2 /2 -√2 /2 ·cos x <=>

sin( x -π/4 ) = √2 /2 ( sin x - cos x ) = 1 <=>

sin x -cox x = √2 <=> ( sin x -cos x )² = 2 <=>

sin²x +cos²x -sin 2x = 2 <=> sin 2x = -1 (deoarece sin²x +cos²x = 1).

Obtinem ca x = 3·π/4 .

Verificare: sin 2·3·π/4 = sin 3·π/2 = sin (2π -π/2) = -sin(π/2) = -1 (ceea ce este adevarat).