7. $$ \frac{2\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)}{3\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)} $$

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UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ \frac{2(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))}{3(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{2\left(\cos\left(\frac{\pi }{2}\right)+i\sin\left(\frac{\pi }{2}\right)\right)}{3\left(\cos\left(\frac{\pi }{3}\right)+i\sin\left(\frac{\pi }{3}\right)\right)}$$
- step1: Using the formula:
  $$\frac{2}{3}\left(\cos\left(\frac{\pi }{2}-\frac{\pi }{3}\right)+i\sin\left(\frac{\pi }{2}-\frac{\pi }{3}\right)\right)$$
- step2: Calculate:
  $$\frac{2}{3}\left(\cos\left(\frac{\pi }{6}\right)+i\sin\left(\frac{\pi }{2}-\frac{\pi }{3}\right)\right)$$
- step3: Calculate:
  $$\frac{2}{3}\left(\cos\left(\frac{\pi }{6}\right)+i\sin\left(\frac{\pi }{6}\right)\right)$$
- step4: Simplify:
  $$\frac{\sqrt{3}}{3}+\frac{1}{3}i$$
เมื่อคำนวณเศษส่วนของมุมที่กำหนด เราจะได้ผลลัพธ์เป็น $$ \frac{\sqrt{3}}{3} + \frac{1}{3}i $$
ผลลัพธ์ของสมการ $$ \frac{2(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))}{3(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))} $$ คือ $$ \frac{\sqrt{3}}{3} + \frac{1}{3}i $$

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