Find (-1+ i)"and write the answer in exact polar form and rectangular form.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
100%
![**Problem Statement:**
Find \(( -1 + i )^6\) and write the answer in exact polar form and rectangular form.
**Solution:**
To solve the given problem, we will convert the complex number \( -1 + i \) to its polar form and then use De Moivre’s Theorem to find the desired power. Finally, we will convert the answer back to rectangular form.
### Step 1: Convert \( -1 + i \) to Polar Form
A complex number \( z \) in rectangular form is expressed as \( z = x + yi \), where \( x \) and \( y \) are real numbers. For \( -1 + i \):
- \( x = -1 \)
- \( y = 1 \)
The polar form is \( r (\cos \theta + i \sin \theta) \), where:
- \( r \) is the modulus of the complex number, found using \( r = \sqrt{x^2 + y^2} \).
- \( \theta \) is the argument (angle) of the complex number, found using \( \tan \theta = \frac{y}{x} \).
Calculate the modulus \( r \):
\[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Calculate the argument \( \theta \):
\[ \tan \theta = \frac{1}{-1} = -1 \]
\[ \theta = \tan^{-1}(-1) \]
Since the complex number \( -1 + i \) is in the second quadrant, we have:
\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]
Therefore, the polar form of \( -1 + i \) is:
\[ \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]
### Step 2: Apply De Moivre's Theorem to Find \(( -1 + i )^6\)
Using De Moivre’s Theorem \( \left( r (\cos \theta + i \sin \theta) \right)^n = r^n (\cos n\theta + i \sin n\theta) \):
Let \( n =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e4013eb-babb-4770-9243-fb3338b3d4a1%2F86253f4c-45f8-4fea-8eed-121f1f90d72a%2Forc5bhr.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \(( -1 + i )^6\) and write the answer in exact polar form and rectangular form.
**Solution:**
To solve the given problem, we will convert the complex number \( -1 + i \) to its polar form and then use De Moivre’s Theorem to find the desired power. Finally, we will convert the answer back to rectangular form.
### Step 1: Convert \( -1 + i \) to Polar Form
A complex number \( z \) in rectangular form is expressed as \( z = x + yi \), where \( x \) and \( y \) are real numbers. For \( -1 + i \):
- \( x = -1 \)
- \( y = 1 \)
The polar form is \( r (\cos \theta + i \sin \theta) \), where:
- \( r \) is the modulus of the complex number, found using \( r = \sqrt{x^2 + y^2} \).
- \( \theta \) is the argument (angle) of the complex number, found using \( \tan \theta = \frac{y}{x} \).
Calculate the modulus \( r \):
\[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Calculate the argument \( \theta \):
\[ \tan \theta = \frac{1}{-1} = -1 \]
\[ \theta = \tan^{-1}(-1) \]
Since the complex number \( -1 + i \) is in the second quadrant, we have:
\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]
Therefore, the polar form of \( -1 + i \) is:
\[ \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]
### Step 2: Apply De Moivre's Theorem to Find \(( -1 + i )^6\)
Using De Moivre’s Theorem \( \left( r (\cos \theta + i \sin \theta) \right)^n = r^n (\cos n\theta + i \sin n\theta) \):
Let \( n =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning