How do I verify that $\sin (\theta)$ and $\cos (\theta)$ are functions? I am studying pre-calculus mathematics at the moment, and I need help in verifying if  $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ and $\cos (\theta)$. How do I go about showing this?

The proof is based simply on similar triangles. If a right-angled triangle has an angle $\theta$ then the other two angles are $90^{\circ}$ and $(90-\theta)^{\circ}$. If two triangles have the same angles then they are similar.
My picture shows two similar triangles: $\triangle OAB$ and $\triangle OA'B'$. 
Since $\theta = \angle AOB$ then, by definition 
$$\sin\theta = \frac{\|AB\|}{\|OB\|}$$
Since $\theta = \angle A'OB'$ then, by definition
$$\sin\theta = \frac{\|A'B'\|}{\|OB'\|}$$
We can show that $\sin \theta$ has a single, unique value if we can show that the two ratios agree.
Let $T$ be the linear transformation given by an enlargement, centre $O$, with scale factor $\lambda$, such that $T(A) = A'$ and $T(B) = B'$. We have $\|A'B'\|=\lambda\|AB\|$ and $\|OB'\| = \lambda\|OB\|$, hence
$$\frac{\|A'B'\|}{\|OB'\|} = \frac{\lambda\|AB\|}{\lambda\|OB\|}=\frac{\|AB\|}{\|OB\|}$$
This shows that the ratio of the opposite side to the hypotenuse is the same for any two similar, right-angled triangles with angle $\theta$. That means that $\sin\theta$ is uniquely, well-defined.
(N.B. Similarity allows rotation and reflection as well as enlargement. However, rotations and reflections preserve lengths and so preserve ratios of lengths.)

