Use Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ and its complex conjugate to find formulas for $\sin\theta$ and $\cos\theta$. In your physics career, you will often need to read these formula “backwards,” i.e. notice one of these combinations of exponentials in a sea of other symbols and say, “Ah ha! that is $\cos\theta$.” So, pay attention to the result of the homework problem!
With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states $\ket{\psi_i}$ $\left( i=1,2,3,4\right)$.
Use your measured probabilities to find each of the unknown states as a linear superposition of the $S_z$-basis states $\ket{+}$ and $\ket{-}$.
Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
Using the Spin 1 version of the Spins simulation, find the probabilities for the results of the three spin components for unknowns $\ket{\psi_1}$ and $\ket{\psi_3}$. Use these probabilities to write the unknowns in the $z$-basis.