1. **Problem Statement:** Use De Moivre's Theorem to find $(\cos \theta + i \sin \theta)^n$ for given $\theta$ and integer $n$. 2. **Formula:** De Moivre's Theorem states that for any real number $\theta$ and integer $n$, $$ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) $$ This theorem helps us raise complex numbers in trigonometric form to integer powers easily. 3. **Important Rules:** - $i$ is the imaginary unit with $i^2 = -1$. - $\cos$ and $\sin$ are trigonometric functions representing the real and imaginary parts respectively. - $n$ must be an integer (positive, negative, or zero). 4. **Example:** Suppose we want to compute $(\cos 30^\circ + i \sin 30^\circ)^3$. - Convert degrees to radians if needed: $30^\circ = \frac{\pi}{6}$. - Apply De Moivre's Theorem: $$ (\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})^3 = \cos(3 \times \frac{\pi}{6}) + i \sin(3 \times \frac{\pi}{6}) = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} $$ - Evaluate: $\cos \frac{\pi}{2} = 0$, $\sin \frac{\pi}{2} = 1$. - So the result is $0 + i \times 1 = i$. 5. **Interpretation:** Raising the complex number on the unit circle to the power $n$ rotates it by $n$ times the angle $\theta$. **Final answer:** $$ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) $$