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Chapter 6: Problem 54

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$\left[2\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^{3}$$

Short Answer

Expert verified
The rectangular form of the complex number \(\[2(\cos 10^{\circ}+i \sin 10^{\circ})\]^{3}\) is \(\4\sqrt{3} + 4i\).

Step by step solution

01

Identify r, 胃 and n

Here we have \(r = 2\), \(\theta = 10^{\circ}\) and \(n = 3\) from the given expression \(\[2(\cos 10^{\circ}+i \sin 10^{\circ})\]^{3}\)
02

Apply DeMoivre's Theorem

By applying DeMoivre's theorem, we get \(\[r^n (\cos n\theta + i \sin n\theta)\] = \[2^3 (\cos 3*10^{\circ} + i \sin 3*10^{\circ})\] = \[8 (\cos 30^{\circ} + i \sin 30^{\circ})\]
03

Convert the Result to Rectangular Form

To convert this to rectangular form, we use the identities \(\cos 30^{\circ} = \sqrt{3}/2\) and \(\sin 30^{\circ} = 1/2\). Thus, \(\cos 30^{\circ} + i \sin 30^{\circ} = \sqrt{3}/2 + i/2\). Multiplying these values by \(8\) (which is our \(r^n\)), we get \(4\sqrt{3} + 4i\) as the rectangular form of the complex number.

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