To solve the expression \( 2 \cdot e^{i \cdot 210^\circ} \), we can use Euler's formula:

\[
e^{i\theta} = \cos\theta + i\sin\theta
\]

### Step 1: Convert the angle to radians
Since Euler's formula works with angles in radians, we need to convert \( 210^\circ \) to radians.

\[
210^\circ = \frac{210 \pi}{180} = \frac{7\pi}{6}
\]

### Step 2: Apply Euler's formula
Using \( \theta = \frac{7\pi}{6} \), we substitute into Euler's formula:

\[
e^{i \cdot \frac{7\pi}{6}} = \cos\left( \frac{7\pi}{6} \right) + i \sin\left( \frac{7\pi}{6} \right)
\]

### Step 3: Find the trigonometric values
The angle \( \frac{7\pi}{6} \) corresponds to \( 210^\circ \), which lies in the third quadrant, where cosine is negative and sine is also negative.

\[
\cos\left( 210^\circ \right) = -\frac{\sqrt{3}}{2}, \quad \sin\left( 210^\circ \right) = -\frac{1}{2}
\]

So,

\[
e^{i \cdot 210^\circ} = -\frac{\sqrt{3}}{2} - i \cdot \frac{1}{2}
\]

### Step 4: Multiply by 2
Now, multiply the result by 2:

\[
2 \cdot \left( -\frac{\sqrt{3}}{2} - i \cdot \frac{1}{2} \right) = -\sqrt{3} - i
\]

### Final Answer:
\[
2 \cdot e^{i \cdot 210^\circ} = -\sqrt{3} - i
\]

Let me know if you would like more details or have questions!

Here are 5 related questions:
1. How does Euler's formula relate to complex numbers?
2. What are the properties of trigonometric functions in different quadrants?
3. Can you explain the significance of multiplying complex numbers by real numbers?
4. How do you convert degrees to radians in general?
5. How does the argument of a complex number affect its value?

**Tip:** Always check in which quadrant an angle lies to correctly determine the signs of the sine and cosine values.

Learn how to solve the complex expression 2[e^(i*210°)] using Euler's formula and trigonometric properties. This detailed guide walks you through converting degrees to radians, applying Euler's identity, and solving for real and imaginary components. Suitable for advanced high school students in Grades 11-12.

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