The expression \(3 \operatorname{cis} \left( \frac{\pi}{4} \right)\) represents a complex number in polar form, where \(3\) is the magnitude (or modulus) and \(\frac{\pi}{4}\) is the angle (or argument) in radians. The notation "cis" stands for \( \cos \theta + i \sin \theta \).

To convert this to standard form:

\[
3 \operatorname{cis} \left( \frac{\pi}{4} \right) = 3 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)
\]

Since \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \):

\[
3 \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = \frac{3 \sqrt{2}}{2} + i \frac{3 \sqrt{2}}{2}
\]

Thus, the complex number in standard form is:

\[
\frac{3 \sqrt{2}}{2} + i \frac{3 \sqrt{2}}{2}
\]

Would you like further details on polar-to-rectangular conversion or any other aspect?

Here are five related questions:
1. How is the magnitude of a complex number determined?
2. What are alternative forms of expressing complex numbers?
3. How do you convert rectangular coordinates to polar coordinates?
4. What are common applications of complex numbers in physics?
5. What is Euler's formula, and how does it relate to "cis" notation?

**Tip:** In polar form, the argument angle is typically expressed in radians for simplicity, especially in higher mathematics and physics.

Learn how to convert the complex number 3 cis (π/4) from polar to rectangular form with step-by-step explanations. This solution covers key concepts in complex numbers, including polar and rectangular forms, trigonometric identities, and the cis notation. Suitable for advanced high school or college-level students.

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