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Chapter 6: Problem 55

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$\left[2\left(\cos 80^{\circ}+i \sin 80^{\circ}\right)\right]^{3}$$

Short Answer

Expert verified
The complex number \[ \left[2\left(\cos 80^{\circ}+i \sin 80^{\circ}\right)\right]^{3} \] in rectangular form is \(-4 - 4i\sqrt{3}\).

Step by step solution

01

Apply De Moivre鈥檚 Theorem

By De Moivre's Theorem, the given complex number power operation becomes \[ \left[2^3(\cos (3*80^{\circ})+i \sin (3*80^{\circ}))\right] \] which simplifies to \[ (8(\cos 240^{\circ}+i \sin 240^{\circ})) \].
02

Convert to Rectangular Form

Using the identities \( \cos 240 = -1/2 \) and \( \sin 240 = -\sqrt{3}/2 \), we substitute these values to the polar form equation. This gives us \( 8[(-1/2) + i(-\sqrt{3}/2)] = -4 -4i\sqrt{3} \).
03

Final Answer in Rectangular Form

Therefore, the complex number \[ \left[2\left(\cos 80^{\circ}+i \sin 80^{\circ}\right)\right]^{3} \] in rectangular form is \(-4 - 4i\sqrt{3}\).

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