Consider the following. Square roots of 4(cos 90° + i sin 90°) 0 + 360°k (a) Use the formula z = Vi(cos 0+360°k to find the indicated roots of the complex number. (Enter your answers in +i sin
Consider the following. Square roots of 4(cos 90° + i sin 90°) 0 + 360°k (a) Use the formula z = Vi(cos 0+360°k to find the indicated roots of the complex number. (Enter your answers in +i sin
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve plz and show which graph is correct
Transcribed Image Text:This image contains instructions and four graphs.
**Instructions:**
- "(c) Represent each of the roots graphically."
**Graphs Explanation:**
Four complex plane graphs are shown with the following features:
1. **Axes**:
- Each graph has a horizontal axis labeled "Real axis" ranging from -10 to 10.
- The vertical axis is labeled "Imaginary axis," also ranging from -10 to 10.
2. **Circle**:
- A circle is centered at the origin (0, 0) in each graph with a radius extending to the 5 on either axis.
3. **Arrows**:
- In each graph, arrows along the circle indicate rotational movement, possibly representing roots of unity or similar concepts in complex numbers.
At the bottom-left of the image, a prompt reads, "Need Help? Read It," perhaps indicating additional resources or guidance is available.
This setup visualizes how complex roots can be represented on the complex plane, highlighting both real and imaginary components.
![**Consider the following.**
Square roots of \(4(\cos 90^\circ + i \sin 90^\circ)\)
(a) Use the formula \(z_k = \sqrt[n]{r}\left(\cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n}\right)\) to find the indicated roots of the complex number. (Enter your answers in trigonometric form. Let \(0 \leq \theta < 360^\circ\).)
\[ z_0 = \]
\[ z_1 = \]
(b) Write each of the roots in standard form.
\[ z_0 = \]
\[ z_1 = \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7aa7f844-e0ae-47d2-8d06-fbc16e2cc765%2Fd6329ca4-2778-4758-b65b-2cb025031714%2Fq27dzwd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the following.**
Square roots of \(4(\cos 90^\circ + i \sin 90^\circ)\)
(a) Use the formula \(z_k = \sqrt[n]{r}\left(\cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n}\right)\) to find the indicated roots of the complex number. (Enter your answers in trigonometric form. Let \(0 \leq \theta < 360^\circ\).)
\[ z_0 = \]
\[ z_1 = \]
(b) Write each of the roots in standard form.
\[ z_0 = \]
\[ z_1 = \]
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