What does being proportional mean?

Question

Question:

Two concurrent forces act along the sides CA and CB of a triangle. Their magnitudes are proportional to $\cos (A)$ and $\cos (B)$ respectively. Prove that their resultant is proportional to $\sin(C)$.

My book's attempt:

Let the forces acting along $CA$ and $CB$ be $\vec{P}$ and $\vec{Q}$ respectively. Let the resultant of $\vec{P}$ and $\vec{Q}$ be $\vec{R}$, which is acting along $CD$. Now, let $P=k\cos(A)$, $Q=k\cos(B)$, where $k$ is the constant of proportionality.

$$....$$

$$[\text{Then my book went on to prove that}\ R=k\sin(C)]$$

My comments:

If $P$ is proportional to $\cos(A)$, it means that $P=n\cos(A)$. Again, if $Q$ is proportional to $\cos(B)$, it means that $Q=s\cos(B)$. Furthermore, if $R$ is proportional to $\sin(C)$, it means that $R=t\sin(C)$. Now, the constants of proportionality $n,s,t$ needn't be equal. Why did my book assume that the constants were equal i.e. why did my book assume:

$$n=s=t=k$$

My question:

1. Why did my book assume that the constants of proportionality were equal? Is that included in the definition of the word 'proportional'?

Answer 1

Question:

Two concurrent forces act along the sides CA and CB of a triangle. Their magnitudes are proportional to $\cos (A)$ and $\cos (B)$ respectively.

My book's attempt:

Let the forces acting along $CA$ and $CB$ be $\vec{P}$ and $\vec{Q}$ respectively. Let the resultant of $\vec{P}$ and $\vec{Q}$ be $\vec{R}.$

Since the magnitudes and the cosines are in direct proportion, each corresponding-magnitude-cosine pair is in the same ratio. That is, $$\frac P{\cos A}=\frac Q{\cos B}.$$

My question:

If $P$ is proportional to $\cos(A)$, it means that $P=n\cos(A)$. Again, if $Q$ is proportional to $\cos(B)$, it means that $Q=s\cos(B)$. Why did my book assume that the constants of proportionality were equal?

Let $m_1,m_2,m_3$ represent the three magnitudes and $c_1,c_2,c_3$ correspondingly represent the three cosines. I.e., $m_1=P, c_1=\cos A,$ etc.

To be clear, the given information is $$m\propto c,\\\text{i.e., }\frac{m_1}{c_1}=k=\frac{m_2}{c_2},$$ rather than $$m_1\propto c_1,\;m_2\propto c_2,\\\text{i.e., }m_1=k_1c_1,\;m_2=k_2c_2.$$

Analogously, if $x=7y,$ then we say that $x$ is proportional to $y;$ we don't say that $21$ is proportional to $3.$

$m$ and $x$ are variables, while $P$ and $21$ are particular values or constants.