Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 5.2.61

Chapter 5, Problem 5.2.61

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 + i)⁵

Verified step by step guidance

Recall DeMoivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), its \(n\)th power is given by \(r^n (\cos(n\theta) + i \sin(n\theta))\).

Convert the complex number \(1 + i\) into polar form by finding its magnitude \(r\) and argument \(\theta\). The magnitude is \(r = \sqrt{1^2 + 1^2}\) and the argument is \(\theta = \arctan\left(\frac{1}{1}\right)\).

Apply DeMoivre's Theorem to raise the complex number to the 5th power: compute \(r^5\) and multiply the angle by 5 to get \(5\theta\). So the result in polar form is \(r^5 (\cos(5\theta) + i \sin(5\theta))\).

Convert the result back to rectangular form by evaluating \(r^5 \cos(5\theta)\) for the real part and \(r^5 \sin(5\theta)\) for the imaginary part.

Write the final answer as \(a + bi\), where \(a\) and \(b\) are the real and imaginary parts found in the previous step.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number expressed in polar form as r(cos θ + i sin θ), its nth power is given by r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to powers by working with their magnitude and angle instead of expanding binomials.

Conversion between Rectangular and Polar Forms

Complex numbers can be represented in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Converting to polar form involves finding the magnitude r = √(a² + b²) and the argument θ = arctan(b/a), which is essential for applying DeMoivre's Theorem.

Rectangular Form of Complex Numbers

Rectangular form expresses complex numbers as a + bi, where a is the real part and b is the imaginary part. After using DeMoivre's Theorem in polar form, the result is converted back to rectangular form by evaluating cos nθ and sin nθ to find the real and imaginary components.

Recommended video:

03:58

Converting Complex Numbers from Polar to Rectangular Form

Related Practice

Textbook Question

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 4 csc θ

views

Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.

√(−8) (√(−3) − √5 )

views

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. 3x + y = 7

Textbook Question

In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 9(cos 30° + i sin 30°)

views

Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 40° + i sin 40°)]³

views

Textbook Question

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/12 + i sin π/12)]⁶

views