In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/10 + i sin π/10)]⁵

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1. Identify the complex number in polar form: \( z = \frac{1}{2} \left( \cos \frac{\pi}{10} + i \sin \frac{\pi}{10} \right) \).
2. Apply DeMoivre's Theorem, which states \( (r (\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \), where \( r = \frac{1}{2} \), \( \theta = \frac{\pi}{10} \), and \( n = 5 \).
3. Calculate \( r^n = \left( \frac{1}{2} \right)^5 \).
4. Determine the new angle: \( n\theta = 5 \times \frac{\pi}{10} \).
5. Express the result in rectangular form: \( r^n (\cos(n\theta) + i \sin(n\theta)) \) and simplify to get the final expression.

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, particularly when using DeMoivre's Theorem. These functions relate the angles in a triangle to the ratios of its sides and are used to express the argument of a complex number, which is necessary for calculating powers and roots in polar form.