0. Exercises

Try these first

1. 1.

   Show that eiπ + 1 = 0 using Euler’s formula.

   Hint

   Remember what Euler's formula states!

2. 2.

   Express cos(3x) in terms of exponential functions.

   Hint

   Recall the definition of cosine in exponential form.

3. 3.

   What does Euler's formula state?

   • e^(ix) = cos(x) + i*sin(x)
   • e^(ix) = cos(x) - i*sin(x)
   • e^(ix) = i*cos(x) + sin(x)

   Hint

   Focus on the relationship between complex numbers and trigonometric functions.

4. 4.

   True or False: The roots of a complex number can be evenly spaced on a circle.

   • True
   • False

   Hint

   Think about how roots are derived and their geometric representation.

5. 5.

   A high-rise building vibrates due to wind forces modeled by the differential equation d²y/dx² + λy = 0. Determine the form of the solution if λ = 16, and explain the physical meaning of the terms.

   Hint

   Identify how roots relate to the oscillation frequency.

6. 6.

   Transform the following expression into exponential form: sin(2x) + cos(2x). What's the geometric interpretation of this expression on the complex plane?

   Hint

   Consider how both sine and cosine contribute to a single complex circle.