Projection of linear combination of real numbers Suppose a real number $\alpha$ is a rational linear combination of two linearly independent real numbers, such as $\pi$ and $\sqrt 3$. For instance, say $$\alpha = \tfrac12\pi + 5\sqrt 3.$$
By a simple linear algebra argument ($\{\pi,\sqrt3\}$ is a $\mathbb Q$-basis), it's clear that the coefficients of $\alpha$ are uniquely determined.

Question: Given $\alpha\in\operatorname{span}_\mathbb Q(\{\pi,\sqrt 3\})$, how do we determine the coefficients of $\pi$ and $\sqrt 3$?

It feels like there is enough information, but at the same time, I imagine that we can find different coefficients which give us numbers arbitrarily close to $\alpha$, so it is important that we have an exact expression for $\alpha$.
(In other words, if we only know that $\alpha\approx 10.23105$, say, I imagine it would be hopeless to expect that we can determine the coefficients, even if we restrict our view to the module $\operatorname{span}_{\mathbb Z}(\{\pi,\sqrt 3\})$.)

The reason I ask this question is because for each $n$, the integral $$I(n)=\int_0^{\pi/3}\sin^{2n}x\,dx$$ is a number of the form $a\pi+b\sqrt 3$. It's easy to obtain the recurrence relation
$$I(n) = \begin{cases} \hfil\frac\pi3\hfil & \text{if $n=0$}\\[3pt]
(1-\tfrac1{2n})\,I(n-1)-\frac1{4n}\big(\tfrac{\sqrt3}2\big)^{2n-1} & \text{otherwise}
\end{cases}$$
for $I(n)$, but I want to be able to compute the coefficients of $\pi$ and $\sqrt 3$ separately (in exact form) using a computer program.

I appreciate any help with solving either of the two problems, I'm assuming a solution to one necessarily sheds light on the other, that's why I haven't posted these as two separate questions.

Here's easy way to determine them:
let $t \in [0..1]$, we define
$$ a_0 = t * \alpha $$
$$ a_1 = (1-t) * \alpha $$
Basically we have $a_0 + a_1 = \alpha$.
Now, after we have $a_0$ and $a_1$, we can calculate co-efficients:
$$c_0 = \frac{a_0}{\pi}$$
$$c_1 = \frac{a_1}{\sqrt{3}}$$.
I have a feeling that the $t$ cannot be calculated from the $\alpha$, given that they are both located in same dimension, basically you need some criteria how $\alpha \in R$ can be divided to two separate dimensions. Thus your end result is large number of possible alternative coefficients. Every choice of $t$ will have different coefficients. Limiting $c_0,c_1 \in R$ to $Q$ sounds kinda difficult.
