The expression $3 \, \text{cis} \left( \frac{\pi}{4} \right)$ represents a complex number in polar form. Here:

• The modulus (or magnitude) of the complex number is $3$.
• The argument (or angle) is $\frac{\pi}{4}$.

Using the polar form of a complex number, we can convert this into rectangular (Cartesian) form as follows:

$3 \, \text{cis} \left( \frac{\pi}{4} \right) = 3 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$

Since $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, we get:

$3 \, \text{cis} \left( \frac{\pi}{4} \right) = 3 \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right)$

Multiplying $3$ by each term, we obtain:

$= \frac{3\sqrt{2}}{2} + i \frac{3\sqrt{2}}{2}$

So, in rectangular form, the complex number is:

$\frac{3\sqrt{2}}{2} + i \frac{3\sqrt{2}}{2}$

Would you like further details or have any questions?

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1. How is the cis form derived from Euler's formula?
2. What are some common uses for complex numbers in polar form?
3. How would the magnitude change if the angle were doubled?
4. What happens if we use a negative angle in the cis form?
5. How can this form be visualized on the complex plane?

Tip: The "cis" form is shorthand for cosine and sine combined and is useful for simplifying multiplication and division of complex numbers in polar form.