Find the quotient, and write it in rectangular form. 8( cos 480° + i sin 480°) 2 (cos 120° + i sin 120°) 8( cos 480° + i sin 480°) 2 ( cos 120° + i sin 120°) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in the form a + bi.)
Find the quotient, and write it in rectangular form. 8( cos 480° + i sin 480°) 2 (cos 120° + i sin 120°) 8( cos 480° + i sin 480°) 2 ( cos 120° + i sin 120°) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in the form a + bi.)
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
![**Problem Statement:**
Find the quotient, and write it in rectangular form.
\[
\frac{8 \left( \cos 480^\circ + i \sin 480^\circ \right)}{2 \left( \cos 120^\circ + i \sin 120^\circ \right)}
\]
---
\[
\frac{8 \left( \cos 480^\circ + i \sin 480^\circ \right)}{2 \left( \cos 120^\circ + i \sin 120^\circ \right)} = \boxed{\phantom{\frac{1}{1}}}
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in the form \(a + bi\).)
**Explanation:**
The problem requires you to simplify the given complex fraction and convert the result to rectangular form, \(a + bi\), where \(a\) and \(b\) are real numbers.
**Steps for solving:**
1. **Calculate \(\cos 480^\circ\) and \(\sin 480^\circ\):**
- \(\cos 480^\circ = \cos(480^\circ - 360^\circ) = \cos 120^\circ = -\frac{1}{2}\)
- \(\sin 480^\circ = \sin(480^\circ - 360^\circ) = \sin 120^\circ = \frac{\sqrt{3}}{2}\)
2. **Substitute these values into the numerator:**
\[
8 (\cos 480^\circ + i \sin 480^\circ) = 8 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 8 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = -4 + 4i\sqrt{3}
\]
3. **Calculate \(\cos 120^\circ\) and \(\sin 120^\circ\):**
- \(\cos 120^\circ = -\frac{1}{2}\)
- \(\sin 120^\circ = \frac{\sqrt{3}}{2}\)
4. **Substitute these values into the denominator:**
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3def50f-0bee-45b6-a067-ec9125eb6917%2Fc9754015-0e13-4885-a0b0-1a388f2b4a1c%2F3xeskx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the quotient, and write it in rectangular form.
\[
\frac{8 \left( \cos 480^\circ + i \sin 480^\circ \right)}{2 \left( \cos 120^\circ + i \sin 120^\circ \right)}
\]
---
\[
\frac{8 \left( \cos 480^\circ + i \sin 480^\circ \right)}{2 \left( \cos 120^\circ + i \sin 120^\circ \right)} = \boxed{\phantom{\frac{1}{1}}}
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in the form \(a + bi\).)
**Explanation:**
The problem requires you to simplify the given complex fraction and convert the result to rectangular form, \(a + bi\), where \(a\) and \(b\) are real numbers.
**Steps for solving:**
1. **Calculate \(\cos 480^\circ\) and \(\sin 480^\circ\):**
- \(\cos 480^\circ = \cos(480^\circ - 360^\circ) = \cos 120^\circ = -\frac{1}{2}\)
- \(\sin 480^\circ = \sin(480^\circ - 360^\circ) = \sin 120^\circ = \frac{\sqrt{3}}{2}\)
2. **Substitute these values into the numerator:**
\[
8 (\cos 480^\circ + i \sin 480^\circ) = 8 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 8 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = -4 + 4i\sqrt{3}
\]
3. **Calculate \(\cos 120^\circ\) and \(\sin 120^\circ\):**
- \(\cos 120^\circ = -\frac{1}{2}\)
- \(\sin 120^\circ = \frac{\sqrt{3}}{2}\)
4. **Substitute these values into the denominator:**
\[
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 1 images
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