In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form.[4(cos 15° + i sin 15°)]³

Verified step by step guidance

1. Identify the complex number in polar form: \( z = 4(\cos 15^\circ + i \sin 15^\circ) \).
2. Apply DeMoivre's Theorem, which states \( [r(\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta)) \), where \( r = 4 \), \( \theta = 15^\circ \), and \( n = 3 \).
3. Calculate the new modulus: \( r^n = 4^3 \).
4. Calculate the new argument: \( n\theta = 3 \times 15^\circ \).
5. Convert the result back to rectangular form using \( r^n (\cos(n\theta) + i \sin(n\theta)) = a + bi \), where \( a = r^n \cos(n\theta) \) and \( b = r^n \sin(n\theta) \).

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 modulus and θ is the argument). 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 when raising a complex number to a power. Mastery of these functions is necessary for accurate calculations.