To express the complex number $\left(\frac{1}{2} + \frac{i\sqrt{3}}{2}\right)$ in polar and exponential form, we'll follow these steps:

1. Polar Form

A complex number $z = x + iy$ can be expressed in polar form as: $z = r \left(\cos\theta + i\sin\theta\right)$ where:

• $r = \sqrt{x^2 + y^2}$ is the modulus (or magnitude) of the complex number.
• $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ is the argument (or angle) of the complex number.

Given $z = \frac{1}{2} + \frac{i\sqrt{3}}{2}$:

• $x = \frac{1}{2}$
• $y = \frac{\sqrt{3}}{2}$

Step 1: Calculate the modulus $r$

$r = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$

Step 2: Calculate the argument $\theta$

$\theta = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right) = \tan^{-1}(\sqrt{3})$

Since $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$, the argument $\theta$ is $\frac{\pi}{3}$.

Polar Form

So, the polar form of the complex number is: $z = 1 \left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ Or simply: $z = \cos\frac{\pi}{3} + i\sin\frac{\pi}{3}$

2. Exponential Form

The exponential form of a complex number is: $z = r e^{i\theta}$ Substituting the values we found: $z = 1 \cdot e^{i\frac{\pi}{3}} = e^{i\frac{\pi}{3}}$

Final Answer

The complex number $\frac{1}{2} + \frac{i\sqrt{3}}{2}$ in polar form is $\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}$ and in exponential form is $e^{i\frac{\pi}{3}}$.

Would you like more details or have any questions?

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Here are some related questions:

1. How do you convert a complex number from polar to rectangular form?
2. What is the significance of the modulus in polar form?
3. How do you add two complex numbers in polar form?
4. Can a complex number have more than one argument $\theta$?
5. How do you find the cube roots of a complex number in exponential form?

Tip: The argument $\theta$ is typically measured in radians when expressing a complex number in polar or exponential form.