Question

Sinπx=1
Cos(πx/4)=0
Cos(πx/7)=-1
Cos(πx/18)=-корень(3)/2
Cos(πx/6)=-0,5

Answer 1

$1) ;; sin(pi x) = 1\\pi x = dfrac{pi}2} + 2pi n\\x = dfrac{1}{2} + 2n, ;;n in Z\\\2) ;;cos(frac{pi x}{4}) = 0\\dfrac{pi x}{4} = dfrac{pi}{2} + pi n\\x = dfrac{pi}{2}cdot dfrac{4}{pi} + pi n cdot dfrac{4}{pi}\\x = 2 + 4n,;; n in Z\\\3);; cos(frac{pi x}{7}) = -1 \\dfrac{pi x}{7} = pi + 2pi n\\x = pi cdot frac{7}{pi} + 2pi n cdot frac{7}{pi}\\x = 7 + 14n, n in Z$

$4) ;; cos(frac{pi x}{18})=-frac{sqrt3}{2}\\frac{pi x}{18}=pmunderbrace{arccosBig(-frac{sqrt3}{2}Big)}_{pi- arccosfrac{sqrt3}{2}}+2pi n\\frac{pi x}{18}=pm (pi-frac{pi}{6})+2pi n\\frac{pi x}{18}=pmfrac{5pi}{6}+2pi n\\x=pmfrac{5pi}{6}cdot frac{18}{pi}+2pi ncdotfrac{18}{pi}\\x=pm15+36n,;;nin Z$

$5);;cos(frac{pi x}{6})=-frac{1}{2}\\frac{pi x}{6}=pm arccos(-frac{1}{2})+2pi n\\frac{pi x}{6}=pm (pi-frac{pi}{3})+2pi n \\x = pm frac{2pi}{3}cdot frac{6}{pi} + 2pi n cdot frac{6}{pi}\\x = pm 4 + 12n,;; n in Z$

Z - это множество целых чисел