You are working with complex numbers in polar form, where the format is $r \angle \theta$, and the operation you're asking about is the addition of two complex numbers with the same magnitude and angle.

Given: $10 \angle 30^\circ + 10 \angle 30^\circ$

Step 1: Convert to Rectangular Form

Each term is in polar form. We will first convert both terms to rectangular form using the conversion: $r \angle \theta = r (\cos \theta + i \sin \theta)$

For each term, the magnitude $r = 10$ and the angle $\theta = 30^\circ$.

So, both terms become: $10 \angle 30^\circ = 10 (\cos 30^\circ + i \sin 30^\circ)$

Using known values for trigonometric functions at $30^\circ$: $\cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}$ Thus: $10 \angle 30^\circ = 10 \left(\frac{\sqrt{3}}{2} + i \cdot \frac{1}{2}\right)$ $10 \angle 30^\circ = 5\sqrt{3} + 5i$

Step 2: Add the Complex Numbers

Now, add the two rectangular forms: $(5\sqrt{3} + 5i) + (5\sqrt{3} + 5i) = 10\sqrt{3} + 10i$

Step 3: Convert Back to Polar Form

We now convert the result back to polar form:

1. Magnitude: The magnitude $r$ is found by: $r = \sqrt{(10\sqrt{3})^2 + (10)^2} = \sqrt{300 + 100} = \sqrt{400} = 20$
2. Angle: The angle $\theta$ is: $\theta = \tan^{-1}\left(\frac{10}{10\sqrt{3}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = 30^\circ$

Final Answer:

The result in polar form is: $20 \angle 30^\circ$

Let me know if you'd like further clarification or have more questions!

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Related Questions:

1. How do you convert from rectangular form to polar form?
2. What happens when the angles of two complex numbers are different in polar addition?
3. How do you multiply complex numbers in polar form?
4. What is the argument of a complex number in polar form, and how is it determined?
5. How would you subtract complex numbers in polar form?

Tip: When adding complex numbers in polar form, they must have the same magnitude and angle to directly sum them in a simple way. Otherwise, converting them to rectangular form is necessary before adding.