Complicated Cosines Guesswork

This problem requires some guesswork, but it mainly comes down to applying the double and triple angle formulas.

Double and Triple Angle Formulas:

$\sin {2a}=2\sin {a}\cos {a}$

$\sin {3a}=3\sin {a}-4\sin^{3} {a}$

Here's the work:

$\sin {7}=0.122$

$\cos {7}=\sqrt {1-\sin^{2} {7}}$

$\cos {7}$ comes close to 1. None of those are near 0.359.

$\sin {14}$ gets close to 0.244 and $\cos {14}$ gets close to 1 also. Still not close to 0.359...

$3\sin {7}=0.366$

$\sin^{3} {7}=0.122^{3}=0.001815948$

$4\sin^{3} {7}=0.007263792$

$\sin {21}=0.359$

$\cos {69}=0.359$

$\arccos {0.359}=69$

Finally! We found $\arccos {0.359}$ !

Therefore, the answer is $\boxed {69}$