solve-sin-pi-x-4-1-2-

Question Number 87737 by M±th+et£s last updated on 05/Apr/20

s o l v e

s i n (\frac{π}{[\frac{[x]}{4}]}) = \frac{1}{2}

Answered by mahdi last updated on 06/Apr/20

u = [\frac{[x]}{4}] \Rightarrow - 1 ⩽ \frac{1}{u} ⩽ 1 \Rightarrow - π ⩽ \frac{π}{u} ⩽ π {u \neq 0 \Rightarrow x \notin [0, 4)}

\sin (\frac{π}{u}) = \frac{1}{2} \Rightarrow {\begin{cases} \frac{π}{u} = \frac{π}{6} + 2 k π \\ \frac{π}{u} = \frac{5 π}{6} + 2 k π \end{cases}

- π ⩽ \frac{π}{u} ⩽ π \Rightarrow \frac{π}{u} = \frac{π}{6} \lor \frac{5 π}{6}

u = 6 \lor \frac{6}{5} \overset{u \in Z}{\Rightarrow} u = 6

[\frac{[x]}{4}] = 6 \Rightarrow 24 ⩽ x < 28

Commented by M±th+et£s last updated on 05/Apr/20

g o d b l e s s y o u s i r

Commented by M±th+et£s last updated on 06/Apr/20

s i r i f - 24 ⩽ x < - 20 \to \to x = - 0.5 n o t 1

s o S {24 ⩽ x < 8}

Commented by mahdi last updated on 06/Apr/20

what ? how [- 24 ⩽ x < - 20 \Rightarrow x = - 0.5] ?

Commented by M±th+et£s last updated on 06/Apr/20

s o r r y i m e a n a n s = - 0.5 i f - 24 ⩽ x < 20

Commented by M±th+et£s last updated on 06/Apr/20

Commented by mahdi last updated on 06/Apr/20

ok ser, i sorry . {\begin{cases} \frac{π}{6} + 2 k π \Rightarrow \dots, \frac{- 11 π}{6}, \frac{π}{6}, \frac{13 π}{6}, \dots \\ \frac{5 π}{6} + 2 k π \Rightarrow \dots, \frac{- 7 π}{6}, \frac{5 π}{6}, \frac{17 π}{6}, \dots \end{cases}

only : \frac{π}{6} and \frac{5 π}{6} \in [- 1, 1]

Commented by M±th+et£s last updated on 06/Apr/20

t h a n k y o u s i r