Trigonometry: Triangles to Functions

We have two angles at $B$ and $C$ that are subtended by the same arc, so the angles at $B$ and $C$ must be equal.
Also we have triangle $ABD$ which has its longest side as the diameter of the circle, meaning angle $BAD$ is a right-angle.
Just as the angles at $B$ and $C$ are the same, then so are the angles $BAC$ and $BDC$. If we label our two different angles $\theta$ and $\alpha$ then we have the following.
From here we can now write all the angles in terms of $\theta$ and $\alpha$ using our knowledge of triangles, to give us;