Question

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sinx=√2/2 x[-3p/2;5p/2]
sinx=√3/2 x[0;2p]
sinx-√3/2=0 x[-2p;3p/2]

Answer 1

$1)quad sinx=frac{sqrt2}{2}\\x=(-1)^{n}frac{pi}{4}+pi n; ,; nin Z\\xin [-frac{3pi }{2},frac{5pi}{2}]; ; Rightarrow ; ; x=-frac{5pi}{4}; ,; frac{pi}{4}; ,; frac{3pi}{4}; ,; frac{9pi}{4}; .\\2)quad sinx=frac{sqrt3}{2}\\x=(-1)^{n}frac{pi}{3}+pi n,; nin Z\\xin [, 0.2pi ]; ; Rightarrow x=frac{pi}{3}; ,; frac{2pi }{3} , .$

$3)quad sinx=-frac{sqrt3}{2}\\x=(-1)^{n+1}frac{pi}{3}+pi n,; nin Z\\xin [, -2pi , frac{3pi }{2} , ]; ; Rightarrow ; ; x=-frac{2pi }{3}; ,; -frac{pi}{3}; ,; frac{4pi }{3}; .$

$4)quad sinx-frac{sqrt3}{2}=0\\sinx=frac{sqrt3}{2}\\x=(-1)^{n}frac{pi}{3}+pi n,nin Z\\xin[-2pi ,frac{3pi}{2}]to x=-frac{5pi}{3},-frac{4pi}{3},frac{pi}{3},frac{2pi}{3}.$