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$\displaystyle \tan \theta = \sqrt{3}-4\sin 24^\circ$
where, $0<\theta < 90^\circ$.
Find the measure of $\theta$ in degrees.

$sin5x$
$= sin5x + sinx - sinx$
$= 2 sin3x cos2x - sinx$
$= 2 (3sinx - 4sin^3 x) (1 - 2sin^2 x) - sinx$
$= 2 (8sin^5 x - 10sin^3 x + 3sinx) - sinx$
$= 16sin^5 x - 20sin^3 x + 5sinx$

Putting x=6, we get:
$sin30= 16sin^5 6 - 20sin^3 6 + 5sin6$
Solving the equation, we get, $sin6=.105...$,so $cos6=\sqrt{1-sin^2 6}=.995...$,$tan 6=\frac{sin6}{cos6}=.105... .....(i)$

Putting x=12, we get:
$sin60= 16sin^5 12 - 20sin^3 12 + 5sin12$
Solving the equation, we get, $sin12=.208...$,so $cos12=\sqrt{1-sin^2 12}=.978...$

So $sin24=2sin12cos12=.407...$

Now,
$tan\theta=\sqrt{3}-4sin24=.105...=tan 6$[from$i$]
So,$\theta=6$

In this solution, you did some pretty ugly calculations, which are very tough to do without a calculator.
For example, how did you solve the equations for $\sin{6^\circ}$ and $\sin{12^\circ}$ ??
Would you please elaborate...?

Binary search.
It took me a long time to calculate.