To solve the expression [tex]\((\cos 60^\circ + i \sin 60^\circ)^6\)[/tex] and write the answer in rectangular notation, follow these steps:
1. Identify the Components:
- The angle [tex]\(\theta\)[/tex] is [tex]\(60^\circ\)[/tex].
- We need to convert this angle to radians for computations: [tex]\( \theta = 60^\circ \)[/tex].
2. Calculate the Real and Imaginary Parts (Cosine and Sine of the Angle):
[tex]\[
\cos 60^\circ = 0.5
\][/tex]
[tex]\[
\sin 60^\circ = \sqrt{3}/2 \approx 0.866
\][/tex]
3. Apply De Moivre’s Theorem:
De Moivre's theorem states that for a complex number in polar form:
[tex]\[
(r (\cos \theta + i \sin \theta))^n = r^n (\cos(n \theta) + i \sin(n \theta))
\][/tex]
In our case, [tex]\(r = 1\)[/tex] (since the modulus of our complex number is 1), [tex]\(\theta = 60^\circ\)[/tex], and [tex]\(n = 6\)[/tex]:
[tex]\[
(\cos 60^\circ + i \sin 60^\circ)^6 = \cos (6 \cdot 60^\circ) + i \sin (6 \cdot 60^\circ)
\][/tex]
4. Calculate the New Angle:
[tex]\[
6 \cdot 60^\circ = 360^\circ
\][/tex]
Reducing [tex]\(360^\circ\)[/tex] modulo [tex]\(360^\circ\)[/tex] gives [tex]\(0^\circ\)[/tex].
5. Evaluate the Cosine and Sine for the New Angle:
[tex]\[
\cos 360^\circ = \cos 0^\circ = 1
\][/tex]
[tex]\[
\sin 360^\circ = \sin 0^\circ = 0
\][/tex]
6. Combine the Results in Rectangular Form:
[tex]\[
\cos(360^\circ) + i \sin(360^\circ) = 1 + 0i = 1
\][/tex]
Therefore, the value of [tex]\((\cos 60^\circ + i \sin 60^\circ)^6\)[/tex] in rectangular form is:
[tex]\[
1 + 0i \quad \text{or simply} \quad 1
\][/tex]
So the final simplified answer in rectangular form is [tex]\( \boxed{1} \)[/tex].
