Find $\operatorname{Re} \left( e^{\frac{i\pi}{3}} + 2e^{\frac{i\pi}{6}} \right)$ .

The key is converting complex exponentials to rectangular form using Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$. Once you expand each term, the real part is just the cosine component—you don't need the imaginary parts at all. Think of it as extracting the horizontal projection of each rotating vector on the complex plane.

The key is converting complex exponentials to rectangular form using Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$. Once you expand each term, the real part is just the cosine component—you don't need the imaginary parts at all. Think of it as extracting the horizontal projection of each rotating vector on the complex plane.