Math Problem Statement
Both of your answers are incorrect. First, simplify the given expression using De Moivre's Theorem. According to De Moivre's Theorem, if
zequals
r e Superscript i theta
is a complex number, then
z Superscript n
equalsr Superscript n Baseline e Superscript i left parenthesis n theta right parenthesis
.
To use De Moivre's Theorem, first convert the complex number to exponential form,
r e Superscript i theta
.
Then, apply De Moivre's Theorem and simplify. Next, to write the given expression in rectangular form,
x plus y i
,
convert the complex number in exponential form to polar form. Then, evaluate the angles and expand. Be sure that
theta
is in the range
0less than or equals
thetaless than2
Solution
Let's solve using De Moivre's Theorem.
Step 1: Convert to Exponential Form
To use De Moivre's Theorem, we need the complex number in polar form .
Compute the modulus :
Compute the argument :
We need to ensure that is in the correct range . Since is in the fourth quadrant, the angle will be negative:
Write the number in exponential form:
Step 2: Apply De Moivre's Theorem
According to De Moivre's Theorem:
Simplifying the modulus and argument:
Thus:
Step 3: Convert to Rectangular Form
To convert the exponential form back to rectangular form, we need to evaluate the trigonometric functions at the angle :
Compute and :
First, find :
Now compute:
Then:
Thus:
Final Rectangular Form:
Thus:
Final Answers:
- Rectangular form:
- Exponential form:
Do you want any further details or have any questions? Here are some more questions you might find helpful:
- How do you derive trigonometric identities for higher multiples of an angle?
- What is the significance of De Moivre's Theorem in complex analysis?
- Can you explain the process of converting from polar to rectangular form in more detail?
- How do you ensure the angle is in the correct range when using inverse trigonometric functions?
- What are the common pitfalls to avoid when dealing with complex numbers in exponential form?
Tip: Always double-check the quadrant of your angle when converting from rectangular to polar form to ensure the correct sign and value of the argument .
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
De Moivre's Theorem
Exponential Form
Polar Form
Rectangular Form
Formulas
De Moivre's Theorem
Theorems
De Moivre's Theorem
Suitable Grade Level
Advanced High School