sin-3pi-2-cos-x-1-2-

Question Number 86111 by john santu last updated on 27/Mar/20

\sin (\frac{3 π}{2} \cos x) = - \frac{1}{2}

Commented by jagoll last updated on 27/Mar/20

\Leftrightarrow \sin (\frac{3 π}{2} \cos x) = \sin (- \frac{π}{6})

\frac{3 π}{2} \cos x = - \frac{π}{6} + 2 k π

\cos x = \frac{2}{3 π} {- \frac{π}{6} + 2 k π}

\cos x = - \frac{1}{9} + \frac{4 k}{3}

x = \cos^{- 1} (\frac{12 k - 1}{9}) + 2 n π

Answered by TANMAY PANACEA. last updated on 27/Mar/20

s i n (\frac{3 π}{2} c o s x) = s i n (π + \frac{π}{6})

\frac{3 π}{2} c o s x = \frac{7 π}{6}

c o s x = \frac{7}{6} \times \frac{2}{3} = \frac{7}{9} = c o s α

x = 2 n π \pm α [α = {c o s}^{- 1} (\frac{7}{9})]

★ s i n (\frac{3 π}{2} c o s x) = - \frac{1}{2} = s i n (\frac{- π}{6})

\frac{3 π}{2} c o s x = \frac{- π}{6} c o s x = \frac{- 1}{9} = c o s β

x = 2 n π \pm β [β = {c o s}^{- 1} (\frac{- 1}{9})]

Answered by mr W last updated on 27/Mar/20

\sin (\frac{3 π}{2} \cos x) = - \frac{1}{2}

\Rightarrow \frac{3 π}{2} \cos x = n π - {(- 1)}^{n} \frac{π}{6}

\Rightarrow \cos x = \frac{2}{3} [n - {(- 1)}^{n} \frac{1}{6}]

\Rightarrow - 1 ⩽ \frac{2}{3} [n - {(- 1)}^{n} \frac{1}{6}] ⩽ 1

\Rightarrow n = - 1, 0, 1

\Rightarrow \cos x = \frac{2}{3} [n - {(- 1)}^{n} \frac{1}{6}] = - \frac{5}{9}, - \frac{1}{9}, \frac{7}{9}

\Rightarrow x = (2 k + 1) π \pm \cos^{- 1} \frac{5}{9}

\Rightarrow x = (2 k + 1) π \pm \cos^{- 1} \frac{1}{9}

\Rightarrow x = 2 k π \pm \cos^{- 1} \frac{7}{9}

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