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Chapter 6: Problem 56

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$\left[2\left(\cos 40^{\circ}+i \sin 40^{\circ}\right)\right]^{3}$$

Short Answer

Expert verified
The result in rectangular form of the operation \([2(\cos 40^{\circ}+i \sin 40^{\circ})]^3\) is \(-4 + 4\sqrt{3}i\).

Step by step solution

01

Parse given complex number

The given number in this exercise is provided in polar form: \(2(cos 40^{\circ}+i sin 40^{\circ})\), where the amplitude \(r\) is 2 and the argument \(\theta\) is 40 degrees.
02

Apply DeMoivre's theorem

The theorem says: \((r(cos\theta + i sin\theta))^n = r^n(cos(n\theta) + i sin (n\theta))\). In this case, \(n = 3\), so we have: \((2(cos 40^{\circ} + i sin 40^{\circ}))^3 = 2^3(cos(3*40^{\circ}) + i sin (3*40^{\circ})) = 8 (cos 120^{\circ} + i sin 120^{\circ})\)
03

Convert back to rectangular form

In the final step, convert the complex number from polar to rectangular form using the formulas: \(x= r cos\theta\) for the real part and \(y= r sin\theta\) for the imaginary part. Here, \(r = 8\) and \(\theta = 120^{\circ}\). Thus, the rectangular form of the number is \(8 cos 120^{\circ} + 8i sin 120^{\circ} = -4 + 4\sqrt{3}i\)

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