To convert the given complex number in polar form $6\left(\cos \frac{3\pi}{4} + i\sin \frac{3\pi}{4}\right)$ to rectangular form, we need to evaluate the trigonometric functions and then multiply by the magnitude.

Given: $z = 6\left(\cos \frac{3\pi}{4} + i\sin \frac{3\pi}{4}\right)$

Step 1: Evaluate $\cos \frac{3\pi}{4}$ and $\sin \frac{3\pi}{4}$

$\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$ $\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}$

Step 2: Substitute these values into the equation

$z = 6\left(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right)$

Step 3: Distribute the magnitude 6

$z = 6 \cdot \left(-\frac{\sqrt{2}}{2}\right) + 6 \cdot \left(i\frac{\sqrt{2}}{2}\right)$ $z = -3\sqrt{2} + 3i\sqrt{2}$

Final Answer

The rectangular form of the given complex number is: $z = -3\sqrt{2} + 3i\sqrt{2}$

Would you like any more details or have further questions?

Related Questions

1. How do you convert from rectangular to polar form?
2. What is the significance of the argument $\theta$ in polar form?
3. How do you find the magnitude of a complex number given in rectangular form?
4. What is Euler's formula and how does it relate to polar and rectangular forms?
5. Can you explain how to visualize complex numbers on the complex plane?
6. How would you convert $5(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3})$ to rectangular form?
7. What is the general formula for converting from polar to rectangular form?
8. How do you add or subtract complex numbers in polar form?

Tip: Remember that the rectangular form of a complex number $z = x + yi$ can be visualized as a point in the complex plane, with $x$ representing the real part and $y$ representing the imaginary part.