Class problem $7\mathrm{.}1($stresses at an orientation, principal stresses and planes $)$: For the given state of stress, determine

$($a$)$ The principal normal stresses

$($b$)$ Orientation of the planes of maximum normal stress

$($c$)$ The principal shear stress

$($d$)$ Orientation of the planes of maximum shear stress

$($e$)$ the normal and shearing stresses after the element shown has been rotated counter clock wise through $30.$

$($f$)$ the normal and shearing stresses after the element shown has been rotated clock wise through $10$