Combining Sine and Cosine

trigonograph-combiningsineandcosine

For \(\theta\) between \(0^\circ\) and \(90^\circ\):

$$\begin{align}
\large p \sin\theta \,+\, q\cos\theta &\large \;=\; r \sin(\,\theta+\phi\,) \\
\large p \cos\theta \,-\, q\sin\theta &\large \;=\; r \cos(\,\theta+\phi\,) \\[6pt]
\text{where}\quad p^2 + q^2 = r^2 &\quad\text{and}\quad \tan\phi = q/p
\end{align}$$

The resemblance to these trigonographs is no coincidence. (Why?)