5.6. De Moivre’s Theorem

Try these first

1. 1.

   Express (cos(30°) + isin(30°))^2 in terms of cosine and sine.

   Hint

   Use the angle doubling property.

2. 2.

   State De Moivre's Theorem.

   Hint

   Recall the formula involving cosine and sine.

3. 3.

   What does De Moivre's Theorem state?

   • (cosθ + isinθ)n = cos(nθ) + isin(nθ)
   • (cosθ + isinθ)n = sin(nθ) + i cos(nθ)
   • (cosθ + isinθ)n = e^(nθ)

   Hint

   Remember the relationship with powers and how angles are manipulated.

4. 4.

   True or False: De Moivre's Theorem applies only to real numbers.

   • True
   • False

   Hint

   Think about where complex numbers come into play.

5. 5.

   Using De Moivre's Theorem, compute (cos(30°) + i*sin(30°))^6 and provide the answer in rectangular form.

   Hint

   Remember to multiply the angle accordingly.

6. 6.

   Find all cube roots of the complex number 8 using De Moivre's Theorem and provide their polar forms.

   Hint

   You can draw the roots on a circle in the complex plane.