How might I read "$\cos\left(\theta\right):\sin\left(\theta\right):1::x:y:r$"?

Question

In the book I'm reading, A Course in Pure Mathematics, the author writes the following when introducing polar coordinates in section 22:

\begin{align}\cos\left(\theta\right):\sin\left(\theta\right):1::x:y:r\end{align} How do I read this in words correctly? I've never seen such notation before. Here is a snippet of the page in the book:

Answer 1

It’s a three-way proportion. The expression

$$a:b:c::x:y:z\tag{1}$$

means that there is a constant $k$ such that $x=ka,y=kb$, and $z=kc$. Here the relationship between $1$ and $r$ means that we must take $k=r$, so that $x=r\cos\theta$ and $y=r\sin\theta$.

Equivalently, $(1)$ says that any two of $a,b$, and $c$ have the same ratio as the corresponding two of $x,y$, and $z$.

Answer 2

Short answer: Can be handled like fractions $$\left. \begin{gathered} \frac{{\cos (\theta )}}{x} = \frac{1}{r} \hfill \\ \frac{{\sin (\theta )}}{y} = \frac{1}{r} \hfill \\ \end{gathered} \right\} \Rightarrow \begin{array}{*{20}{c}} {x = r\cos (\theta )} \\ {y = r\sin (\theta )} \end{array}$$

Answer 3

When literally read in words, it's "cosine theta is to sine theta is to one as x is to y is to r". The other posts explain the meaning.