  The expectation values of operators drawn from a single quantum state cannot
be outside of a particular region, called their allowed region or the joint
numerical range of the operators. Basically, the allowed region is an image of
the state space under the Born rule. The maximum-eigenvalue-states---of every
linear combination of the operators of interest---are sufficient to generate
boundary of the allowed region. In this way, we obtain the numerical range of
certain Hermitian operators (observables) that are functions of the angular
momentum operators. Especially, we consider here three kinds of
functions---combinations of powers of the ladder operators, powers of the
angular momentum operators and their anticommutators---and discover the allowed
regions of different shapes. By defining some specific concave (and convex)
functions on the joint numerical range, we also achieve tight uncertainty (and
certainty) relations for the observables. Overall, we demonstrate how the
numerical range and uncertainty relations change as the angular momentum
quantum number grows. Finally, we apply the quantum de Finetti theorem by
taking a multi-qubit system and attain the allowed regions and tight
uncertainty relations in the limit where the quantum number goes to infinity.
