1. The problem is to express the complex number $Z=5(\cos(325^\circ) + i\sin(325^\circ))$ in the form $Z = a + bi$ where $a$ and $b$ are real numbers. 2. We use Euler's formula and trigonometric identities. The form $Z = r(\cos \theta + i \sin \theta)$ can be converted to $Z = a + bi$ by calculating $a = r \cos \theta$ and $b = r \sin \theta$. 3. Calculate $a = 5 \cos 325^\circ$ and $b = 5 \sin 325^\circ$. 4. Since $325^\circ = 360^\circ - 35^\circ$, we use the identities: $$\cos 325^\circ = \cos(360^\circ - 35^\circ) = \cos 35^\circ$$ $$\sin 325^\circ = \sin(360^\circ - 35^\circ) = -\sin 35^\circ$$ 5. Using approximate values: $$\cos 35^\circ \approx 0.8192$$ $$\sin 35^\circ \approx 0.5740$$ 6. Therefore: $$a = 5 \times 0.8192 = 4.096$$ $$b = 5 \times (-0.5740) = -2.870$$ 7. The complex number in rectangular form is: $$Z = 4.096 - 2.870i$$