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.
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.
**Rationalize denominators when applicable, DO NOT give any calculator value!!!**
Transcribed Image Text:**Rationalize denominators when applicable, DO NOT give any calculator value!!!**
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