Find the indicated roots. Express the results in rectangular...

Find the indicated roots. Express the results in rectangular form.
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$1+\cos \theta+i \sin \theta=2 \cos \left(\frac{\theta}{2}\right)\left[\cos \left(\frac{\theta}{2}\right)+i \sin \left(\frac{\theta}{2}\right)\right]$

Key Concepts

Trigonometric Identities

Trigonometric identities, like the half-angle formulas, are equations involving trigonometric functions that hold true for all values of the variables involved. They are used to simplify complex trigonometric expressions and facilitate the conversion of such expressions, as seen in the derivation involving cos(?/2) and sin(?/2).

Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. They provide a way to extend the real number system and are essential for solving equations that have no real solutions. This concept underlies many operations such as addition, multiplication, and extracting roots.

Rectangular Form

The rectangular form of a complex number expresses it as a + bi, making it easy to identify the real and imaginary parts. This form is particularly useful for addition and subtraction, as well as for representing the answer in a manner consistent with standard algebraic notation.

Polar Form Representation

The polar form expresses a complex number in terms of its magnitude (also known as modulus) and an angle (often called the argument or phase). This form is typically written as r(cos ? + i sin ?) and is especially useful for multiplication, division, and finding roots of complex numbers, as it simplifies these operations into manipulations of r and ?.

Euler's Formula

Euler's formula, which states that e^(i?) = cos ? + i sin ?, bridges the concepts of exponential functions and trigonometric functions. This formula is instrumental in moving between the rectangular and polar forms of complex numbers and provides a compact way to work with trigonometric expressions.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form and any integer n, (r(cos ? + i sin ?))^n = r^n (cos(n?) + i sin(n?)). This theorem is crucial for finding the powers and roots of complex numbers, as it simplifies the computations by dealing with the modulus and argument directly.

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