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: \( z = 2(\cos 40^\circ + i \sin 40^\circ) \).
2. Apply DeMoivre's Theorem, which states \( [r(\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta)) \).
3. Calculate the magnitude raised to the power: \( r^3 = 2^3 \).
4. Multiply the angle by the power: \( n\theta = 3 \times 40^\circ \).
5. Convert the result back to rectangular form using \( r^3(\cos(n\theta) + i \sin(n\theta)) \) and evaluate the trigonometric functions.

Key Concepts

DeMoivre's Theorem

DeMoivre's Theorem states that for any complex number expressed in polar form as r(cos θ + i sin θ), the nth power of this complex number can be calculated as r^n (cos(nθ) + i sin(nθ)). This theorem simplifies the process of raising complex numbers to powers and is essential for solving problems involving complex exponentiation.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be represented in two forms: rectangular form (a + bi, where a and b are real numbers) and polar form (r(cos θ + i sin θ), where r is the magnitude and θ is the angle). Understanding how to convert between these forms is crucial for applying DeMoivre's Theorem and expressing the final answer correctly.

Trigonometric Functions

Trigonometric functions such as sine and cosine are fundamental in the context of complex numbers in polar form. They relate the angles and sides of triangles and are used in DeMoivre's Theorem to compute the cosine and sine of the multiplied angle. A solid grasp of these functions is necessary to accurately perform calculations involving angles in complex number problems.