4. Express the following complex number in rectangular form WITHOUT the use of a calculator: \( \sqrt{2}\left(\right. \text{ \cos } \left.60^{\circ}+i \sin 60^{\circ}\right) \)

To express the complex number \( \sqrt{2}\left(\cos 60^{\circ} + i \sin 60^{\circ}\right) \) in rectangular form (which is \( a + ib \)), follow these steps: 1. **Recall the Values of Trigonometric Functions:** - \( \cos 60^{\circ} = \frac{1}{2} \) - \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) 2. **Multiply by \( \sqrt{2} \):** - Real part: \( \sqrt{2} \times \cos 60^{\circ} = \sqrt{2} \times \frac{1}{2} = \frac{\sqrt{2}}{2} \) - Imaginary part: \( \sqrt{2} \times \sin 60^{\circ} = \sqrt{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \) 3. **Combine the Real and Imaginary Parts:** \[ \frac{\sqrt{2}}{2} + i \frac{\sqrt{6}}{2} \] Alternatively, you can factor out \( \frac{1}{2} \): \[ \frac{\sqrt{2} + i\sqrt{6}}{2} \] **Final Answer:** \[ \frac{\sqrt{2}}{2} \,+\, i\,\frac{\sqrt{6}}{2} \]