\( 4 \cos 60 ^ { \circ } + 4 \sin 60 ^ { \circ } i \)

The expression \( 4 \cos 60^{\circ} + 4 \sin 60^{\circ} i \) simplifies beautifully using trigonometric values. Since \( \cos 60^{\circ} = \frac{1}{2} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), plugging these in gives us \( 4 \times \frac{1}{2} + 4 \times \frac{\sqrt{3}}{2} i = 2 + 2\sqrt{3} i \). So, the final result is \( 2 + 2\sqrt{3} i \). This expression is a representation of a complex number in polar form, where the real part is 2 and the imaginary part is \( 2\sqrt{3} \). Understanding complex numbers is essential in various fields such as electrical engineering, where they simplify calculations involving waves and oscillations. Imagine being able to represent electricity’s alternating current effortlessly—now that’s pretty shocking!