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Chapter 6: Problem 60

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$\left[\sqrt{3}\left(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}\right)\right]^{6}$$

Short Answer

Expert verified
The power of the given complex number in rectangular form is \(-13.5 + 23.38i\).

Step by step solution

01

Applying DeMoivre's Theorem

According to DeMoivre's Theorem, for any real number n, \((r(\cos 胃+ i \sin 胃))^n= r^n (\cos n胃 + i \sin n胃)\). Thus, \([\sqrt{3}(\cos \frac{5\pi}{18}+i \sin \frac{5\pi}{18})]^6 = [\sqrt{3}]^6(\cos \frac{6*5\pi}{18}+i \sin\frac{6*5\pi}{18})\).
02

Simplify the Expression

From step 1, [\sqrt{3}]^6 becomes 27, and since \(\cos(2\pi k + \theta) = \cos \theta\) and \(\sin(2\pi k + \theta) = \sin \theta\) for any integer k, \(6 * \frac{5\pi}{18} = \frac{5\pi}{3}\) simplifies to \(\frac{2\pi}{3} + 2\pi\). The expression can now be written as \(27(\cos \frac{2\pi}{3}+i \sin\frac{2\pi}{3})\).
03

Convert to Rectangular Form

The rectangular form of a complex number is represented as \(z = r \cdot(\cos 胃 + i \sin 胃) = r\cos 胃 + r\sin 胃*i\). From step 2, we have \(27(\cos \frac{2\pi}{3}+i \sin\frac{2\pi}{3}) = 27*\cos \frac{2\pi}{3} + 27*\sin \frac{2\pi}{3}*i\). Thus the rectangular form is \(-13.5 + 23.38i\) after evaluating the trigonometric expressions.

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