Find the principal value of each of the following : cos^-1(cos 7π/6)

Question

Find the principal value of each of the following :

\(\cos^{-1}\left(\cos\frac{7\pi}{6}\right)\)

Answer 1

\(\cos^{-1}\left(\cos\frac{7\pi}{6}\right)\)\(=\cos^{-1}\left(\cos\left(2\pi+\frac{5\pi}{6}\right)\right)\)

[Formula: cos(2π – x) = cos (x), as cos has a positive vaule in the fourth quadrant. ]

\(=\cos^{-1}\left(\cos\frac{5\pi}{6}\right)\) [Formula: cos ^-1(cos x) = x

\(=\frac{5\pi}{6}\)