**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 FormA 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 =

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Trigonometry

Find (-1+ i)"and write the answer in exact polar form and rectangular form.

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Complex Numbers

Section: Chapter Questions

Problem 7T

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.

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**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 =

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hey can you help me with this please?
Answers are attached, I'm confused about part B. Cos x/r = -8/radical 145 and Sin y/r = 9/radical 145, but they don't produce the same answers? Both answers are between 0 to 2pi. You can also use tan y/x = 9/-8 right?