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.

Using Mathematica
RSolve[{i[n] == (1 - 1/(2 n)) i[n - 1] - 
    1/(4 n) (Sqrt[3]/2)^(2 n - 1), i[0] == Pi/3}, i[n], n]

$$I_n=\frac{3^{n+\frac{1}{2}} \, _2F_1\left(1,n+1;n+\frac{3}{2};\frac{3}{4}\right)}{2^{2 n+2}(2 n+1) }$$
Where $_2F_1$ is the  hypergeometric function and has the series expansion
$$_2F_1(a,b;c;z)=\sum _{k=0}^{\infty } \frac{a_k b_k z^k}{k! c_k}$$
$a_k,b_k,c_k$ are the Pochhammer symbol 
In the table below the first exact values of the integrals

$$
\begin{array}{c|r}
 n & I_n\\
\hline
 0 & \frac{\pi }{3} \\
 1 & \frac{4 \sqrt{3} \pi -9}{24 \sqrt{3}} \\
 2 & \frac{8 \sqrt{3} \pi -27}{64 \sqrt{3}} \\
 3 & \frac{20 \sqrt{3} \pi -81}{192 \sqrt{3}} \\
 4 & \frac{560 \sqrt{3} \pi -2511}{6144 \sqrt{3}} \\
 5 & \frac{\sqrt{3} \left(560 \sqrt{3} \pi -2673\right)}{20480} \\
 6 & \frac{7 \left(440 \sqrt{3} \pi -2187\right)}{40960 \sqrt{3}} \\
\ldots & \ldots\\
\end{array}
$$
