In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 40° + i sin 40°)]³

Verified step by step guidance

1. Identify the complex number in polar form: \(2(\cos 40^\circ + i \sin 40^\circ)\), where the modulus \(r = 2\) and the argument \(\theta = 40^\circ\).
2. Recall DeMoivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), its \(n\)th power is given by \(r^n (\cos n\theta + i \sin n\theta)\).
3. Apply DeMoivre's Theorem with \(n = 3\): compute the new modulus as \(r^3 = 2^3\) and the new argument as \(3 \times 40^\circ\).
4. Write the resulting complex number in polar form: \(2^3 (\cos 120^\circ + i \sin 120^\circ)\).
5. Convert the polar form back to rectangular form by calculating \(2^3 \cos 120^\circ\) for the real part and \(2^3 \sin 120^\circ\) for the imaginary part, then express the answer as \(a + bi\).

Key Concepts

DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). It allows raising complex numbers to integer powers by multiplying the angle and raising the magnitude to the power.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ) or in rectangular form as a + bi. Converting between these forms is essential, especially after applying DeMoivre's Theorem, to write the final answer in standard a + bi format.

Trigonometric Identities for Powers and Angles

Understanding how to compute cos(nθ) and sin(nθ) for multiples of angles is crucial. This involves using angle multiplication and sometimes trigonometric identities to simplify expressions when applying DeMoivre's Theorem.