use de moivres theorem to expand cos^4θ in terms of multiple powers of z based on θ

To expand \( \cos^4 \theta \) using De Moivre's theorem, we express powers of \( \cos \theta \) in terms of complex exponentials. First, let's recall De Moivre's theorem:

\[
\left( \cos \theta + i \sin \theta \right)^n = \cos (n\theta) + i \sin (n\theta).
\]

Now, introduce \( z = e^{i\theta} \), where:

\[
z = \cos \theta + i \sin \theta \quad \text{and} \quad z^{-1} = \cos \theta - i \sin \theta.
\]

From this, we know:

\[
\cos \theta = \frac{z + z^{-1}}{2}.
\]

### Step 1: Express \( \cos^4 \theta \) in terms of \( z \)
Now let's express \( \cos^4 \theta \) using this identity:

\[
\cos^4 \theta = \left( \frac{z + z^{-1}}{2} \right)^4.
\]

### Step 2: Expand the expression
Expand the binomial:

\[
\cos^4 \theta = \frac{1}{16} \left( z + z^{-1} \right)^4.
\]

Using the binomial expansion:

\[
\left( z + z^{-1} \right)^4 = z^4 + 4z^3z^{-1} + 6z^2z^{-2} + 4z z^{-3} + z^{-4}.
\]

Simplify the powers of \( z \):

\[
\left( z + z^{-1} \right)^4 = z^4 + 4z^2 + 6 + 4z^{-2} + z^{-4}.
\]

### Step 3: Combine terms
Now we substitute this back into the expression for \( \cos^4 \theta \):

\[
\cos^4 \theta = \frac{1}{16} \left( z^4 + 4z^2 + 6 + 4z^{-2} + z^{-4} \right).
\]

### Final Answer:
So, the expansion of \( \cos^4 \theta \) in terms of powers of \( z \) is:

\[
\cos^4 \theta = \frac{1}{16} \left( z^4 + 4z^2 + 6 + 4z^{-2} + z^{-4} \right).
\]

Would you like further details or have any questions about this? 

Here are 5 related questions:
1. How can De Moivre's theorem be applied to express \( \sin^4 \theta \)?
2. Can you derive the expression for \( \cos^2 \theta \) using De Moivre's theorem?
3. How does Euler's formula relate to De Moivre's theorem?
4. What happens when expanding \( \cos^6 \theta \) in terms of powers of \( z \)?
5. How can you use De Moivre's theorem to compute \( \cos(4\theta) \)?

**Tip:** De Moivre's theorem is especially useful when dealing with powers of trigonometric functions.

Learn how to expand cos^4θ using De Moivre's Theorem and complex exponentials in terms of powers of z. This detailed solution covers key trigonometric and exponential concepts, making it suitable for advanced high school or undergraduate math students.

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