Question
4. Express the following complex number in rectangular form WITHOUT the use of a calculator: \( \sqrt{2}\left(\right. \text{ \cos } \left.60^{\circ}+i \sin 60^{\circ}\right) \)
Ask by Salazar Tucker. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The complex number in rectangular form is \( \frac{\sqrt{2}}{2} + i \frac{\sqrt{6}}{2} \).
Solution
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:**
\[
\frac{\sqrt{2}}{2} \,+\, i\,\frac{\sqrt{6}}{2}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To express the complex number in rectangular form, we start with the given expression: \[ \sqrt{2}\left( \cos 60^{\circ} + i \sin 60^{\circ} \right) \] From trigonometric values, we know that: \[ \cos 60^{\circ} = \frac{1}{2} \quad \text{and} \quad \sin 60^{\circ} = \frac{\sqrt{3}}{2} \] Substituting these values in, we have: \[ \sqrt{2}\left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) \] Next, we distribute \(\sqrt{2}\): \[ \sqrt{2} \cdot \frac{1}{2} + i \sqrt{2} \cdot \frac{\sqrt{3}}{2} \] This simplifies to: \[ \frac{\sqrt{2}}{2} + i \frac{\sqrt{6}}{2} \] Thus, the rectangular form of the complex number is: \[ \frac{\sqrt{2}}{2} + i \frac{\sqrt{6}}{2} \]
