In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 80° + i sin 80°)]³

Verified step by step guidance

1. Identify the complex number in polar form: $z=2(\cos {80}^{\circ }+i\sin {80}^{\circ })$.
2. Apply DeMoivre's Theorem: $z^n=r^n(\cos (n\theta )+i\sin (n\theta ))$, where $r=2$, $\theta ={80}^{\circ }$, and $n=3$.
3. Calculate the magnitude: $r^n=2^3=8$.
4. Calculate the new angle: $n\theta =3\times {80}^{\circ }={240}^{\circ }$.
5. Convert back to rectangular form: $8(\cos {240}^{\circ }+i\sin {240}^{\circ })$ and use $\cos {240}^{\circ }=-\frac{1}{2}$ and $\sin {240}^{\circ }=-\frac{\sqrt{3}}{2}$ to express in $a+bi$ form.

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 fundamental in connecting trigonometric functions with complex numbers.

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 essential for applying DeMoivre's Theorem and for expressing the final result in the required rectangular form.

Trigonometric Functions

Trigonometric functions such as sine and cosine are crucial in the context of complex numbers, particularly when using DeMoivre's Theorem. These functions relate the angles and sides of triangles and are used to express the argument of a complex number. Familiarity with their values and properties is necessary for accurately calculating powers and converting results back to rectangular form.