Certainly! To simplify the expression [tex]\((\cos 2\theta + i \sin 2\theta)(\cos 5\theta + i \sin 5\theta)\)[/tex] using de Moivre's theorem, follow these steps:
1. Identify the Angles:
- Let [tex]\(a = 2\theta\)[/tex].
- Let [tex]\(b = 5\theta\)[/tex].
2. Apply de Moivre's Theorem:
de Moivre's theorem states that for any two complex numbers in polar form,
[tex]\[
(\cos a + i \sin a)(\cos b + i \sin b) = \cos(a + b) + i\sin(a + b).
\][/tex]
Here, [tex]\(a = 2\theta\)[/tex] and [tex]\(b = 5\theta\)[/tex]. So, the expression becomes:
[tex]\[
(\cos 2\theta + i \sin 2\theta)(\cos 5\theta + i \sin 5\theta) = \cos(2\theta + 5\theta) + i\sin(2\theta + 5\theta).
\][/tex]
3. Simplify the Angles Inside the Cosine and Sine:
- Calculate the sum of the angles:
[tex]\[
2\theta + 5\theta = 7\theta.
\][/tex]
4. Resulting Expression:
Thus, the simplified form of the given expression using de Moivre's theorem is:
[tex]\[
\cos 7\theta + i \sin 7\theta.
\][/tex]
5. Evaluate for a Specific Angle (If Needed):
Let's consider that we need to evaluate [tex]\(\cos 7\theta\)[/tex] and [tex]\(\sin 7\theta\)[/tex] for a particular angle [tex]\(\theta\)[/tex]. Suppose we have a specific value of [tex]\(\theta\)[/tex], for example, [tex]\(\theta = 1\)[/tex]. Calculate:
[tex]\[
7\theta = 7 \times 1 = 7.
\][/tex]
Then,
[tex]\[
\cos 7 \quad \text{and} \quad \sin 7.
\][/tex]
For these angles, the corresponding values might be:
[tex]\[
\cos 7 \approx 0.7539022543433046,
\][/tex]
[tex]\[
\sin 7 \approx 0.6569865987187891.
\][/tex]
So, the expression [tex]\(\cos 7\theta + i \sin 7\theta\)[/tex] evaluates to approximately:
[tex]\[
0.7539022543433046 + i \times 0.6569865987187891.
\][/tex]
Hence, the simplified expression using de Moivre's theorem is:
[tex]\[
\boxed{\cos 7\theta + i \sin 7\theta}
\][/tex]
