Question

Sa se calculeze arcsin(sin7).
Va rog explicație?!!?

Answer 1

$\boxed{arcin(\sin x)=x, \ x \in [\frac{-\pi}{2}; \ \frac{\pi}{2}]}\\ \\ 7=2\pi+7-2\pi\\ \\ \Rightarrow \sin 7=\sin(2\pi+7-2\pi)=\sin(7-2\pi)\\ \\ 2\pi \ aprox. \ 6,28\\ \\ \Rightarrow 7-2\pi \ aprox. \ 0,72\\ \\ 0<0,72<\frac{\pi}{2} \ (aprox. \ 1,57)\\ \\ \Rightarrow arcsin(\sin 7)=arcsin(\sin (7-2\pi)), \ 7-2\pi \in [\frac{-\pi}{2}; \ \frac{\pi}{2}]\\ \\ \Rightarrow arcsin(\sin (7-2\pi))=7-2\pi$