Finding the Real Part of $6e^{i\pi/3}$

To find the real part of the complex number $6e^{i\pi/3}$, we use Euler's formula and properties of complex exponentials.

Step 1: Apply Euler's Formula

Euler's formula states that $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. Applying this to the given expression with $\theta = \pi/3$:

\( e^{i\pi/3} = \cos(\pi/3) + i\sin(\pi/3) \)

Step 2: Evaluate Trigonometric Functions

We know the values for the cosine and sine of $\pi/3$:

• \( \cos(\pi/3) = \frac{1}{2} \)
• \( \sin(\pi/3) = \frac{\sqrt{3}}{2} \)

Substituting these values back:

\( e^{i\pi/3} = \frac{1}{2} + i\frac{\sqrt{3}}{2} \)

Step 3: Multiply by the Magnitude

Now, multiply the result by the scalar part of the expression, which is 6:

\( 6e^{i\pi/3} = 6 \left( \frac{1}{2} + i\frac{\sqrt{3}}{2} \right) \)

Step 4: Distribute and Identify the Real Part

Distribute the 6:

\( 6e^{i\pi/3} = 6 \times \frac{1}{2} + i \left( 6 \times \frac{\sqrt{3}}{2} \right) \)

\( 6e^{i\pi/3} = 3 + i(3\sqrt{3}) \)

The complex number is in the form $a + ib$, where $a$ is the real part and $b$ is the imaginary part. Therefore, the real part is 3.

Conclusion

The real part of $6e^{i\pi/3}$ is 3. This value lies between 2.99 and 3.01, matching the provided answer range.