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Sagot :
[tex]\frac{3\left(-1\right)^{\frac{5}{12}}}{2}[/tex]
1) Let's simplify this expression considering the trigonometric ratios and the complex numbers as well.
[tex]\begin{gathered} 3\left[\cos \left(60^{\circ \:}\right)+i\sin \left(60^{\circ \:}\right)\right]\frac{1}{2}\left[\cos \left(15^{\circ \:}\right)+i\sin \left(15^{\circ \:}\right)\right] \\ Convert\:to\:radians: \\ 3\left[\cos \left(\frac{\pi }{3}\right)+i\sin \left(\frac{\pi }{3}\right)\right]\frac{1}{2}\left[\cos \left(\frac{\pi }{12}\right)+i\sin \left(\frac{\pi }{12}\right)\right] \\ \quad \cos \left(x\right)+i\sin \left(x\right)=e^{ix} \\ 3\times\frac{1}{2}\lbrack\left[e^{i\frac{\pi}{3}}\right]\left[e^{i\frac{\pi}{12}}\right] \\ \frac{3\left(-1\right)^{\frac{5}{12}}}{2} \\ \end{gathered}[/tex]We have transitioned that to work with radians for convenience and used one identity. Note that we could have written our final answer in a radical form.
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