In Exercises 5–7, perform the indicated operation. Leave answers in polar form.[2(cos 10° + i sin 10°)]⁵

Verified step by step guidance

1. Recognize that the expression is in polar form, specifically in the form of \(r(\cos \theta + i \sin \theta)\), where \(r = 2\) and \(\theta = 10^\circ\).
2. Apply De Moivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), the \(n\)-th power is given by \(r^n(\cos(n\theta) + i \sin(n\theta))\).
3. Substitute \(r = 2\), \(\theta = 10^\circ\), and \(n = 5\) into De Moivre's Theorem to get \$2^5(\cos(5 \times 10^\circ) + i \sin(5 \times 10^\circ))$.
4. Calculate \$2^5$ to find the new magnitude of the complex number.
5. Multiply \$5\( by \)10^\circ\( to find the new angle in the polar form, which is \)50^\circ$.

Key Concepts

Polar Form of Complex Numbers

Polar form expresses complex numbers in terms of their magnitude and angle, represented as r(cos θ + i sin θ), where r is the modulus and θ is the argument. This form is particularly useful for multiplication and exponentiation of complex numbers, as it simplifies calculations by allowing the use of trigonometric identities.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the nth power can be calculated as r^n(cos(nθ) + i sin(nθ)). This theorem is essential for raising complex numbers to a power, as it provides a straightforward method to compute the resulting magnitude and angle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. Key identities, such as the sine and cosine addition formulas, are often used in conjunction with polar forms to simplify expressions and solve problems involving angles, especially when performing operations like addition or multiplication of complex numbers.