9. Write in trigonometric form 4(V3 + i)
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°.
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![### Problem 9: Write in Trigonometric Form
Given the complex number \( 4(\sqrt{3} + i) \), write it in trigonometric form.
1. **First, identify the real and imaginary parts:**
- Real part (a): \( 4\sqrt{3} \)
- Imaginary part (b): \( 4 \)
2. **Calculate the magnitude of the complex number:**
Magnitude \( r \) is given by:
\[
r = \sqrt{a^2 + b^2}
\]
Substituting our values,
\[
r = \sqrt{(4\sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8
\]
3. **Determine the argument \( \theta \):**
The argument \( \theta \) (in radians) can be found using the formula:
\[
\theta = \arctan\left(\frac{b}{a}\right)
\]
Substituting our values,
\[
\theta = \arctan\left(\frac{4}{4\sqrt{3}}\right) = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}
\]
4. **Write in trigonometric form:**
The trigonometric form of a complex number is given by:
\[
z = r(\cos \theta + i \sin \theta)
\]
Substituting our values,
\[
4(\sqrt{3} + i) = 8 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)
\]
Thus, the complex number \( 4(\sqrt{3} + i) \) in trigonometric form is:
\[
8 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right).
\]
This concludes the solution for converting the given complex number into its trigonometric form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f9e96de-8215-4532-a7d0-6f1fdf87f241%2F982d87d8-ab4b-4cdd-b508-ad1cd0e5b80a%2Fm8olvpf_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 9: Write in Trigonometric Form
Given the complex number \( 4(\sqrt{3} + i) \), write it in trigonometric form.
1. **First, identify the real and imaginary parts:**
- Real part (a): \( 4\sqrt{3} \)
- Imaginary part (b): \( 4 \)
2. **Calculate the magnitude of the complex number:**
Magnitude \( r \) is given by:
\[
r = \sqrt{a^2 + b^2}
\]
Substituting our values,
\[
r = \sqrt{(4\sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8
\]
3. **Determine the argument \( \theta \):**
The argument \( \theta \) (in radians) can be found using the formula:
\[
\theta = \arctan\left(\frac{b}{a}\right)
\]
Substituting our values,
\[
\theta = \arctan\left(\frac{4}{4\sqrt{3}}\right) = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}
\]
4. **Write in trigonometric form:**
The trigonometric form of a complex number is given by:
\[
z = r(\cos \theta + i \sin \theta)
\]
Substituting our values,
\[
4(\sqrt{3} + i) = 8 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)
\]
Thus, the complex number \( 4(\sqrt{3} + i) \) in trigonometric form is:
\[
8 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right).
\]
This concludes the solution for converting the given complex number into its trigonometric form.
Transcribed Image Text:**Rationalize denominators when applicable, DO NOT give any calculator value!!!**
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