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# Find principal value of the expression $cos^{-1}(cos(-680^{\circ})).$

$680^{\circ}$ is not in the principal interval of cos which is $[0,180^{\circ}]$
Therefore reduce the angle within the principal interval using the result
$cos(n\times\:360^{\circ}-\theta)=cos\theta$
$cos^{-1}cos( 720^{\circ}-40^{\circ})=cos^{-1}cos(2\times\:360^{\circ}-40^{\circ})$
=$cos^{-1}cos40^{\circ}=40^{\circ}$

edited Mar 18, 2013