5.4. Relationship with Trigonometric Functions

Try these first

1. 1.

   Using Euler's identity, express cos(0) in terms of e^(ix).

   Hint

   Remember that e^(0) = 1.

2. 2.

   What is the relationship between sin(x) and e^(ix)?

   Hint

   Focus on the difference of two exponentials.

3. 3.

   What does Euler's identity express?

   • The relationship between triangles
   • The relationship between complex exponentials and trigonometric functions
   • A method for calculus operations

   Hint

   Think about the connection between different mathematical representations.

4. 4.

   True or False: The sine function can be written as a difference of exponentials.

   • True
   • False

   Hint

   Review the definition of sine from a previous lecture.

5. 5.

   Prove that e^(ix) + e^(-ix) = 2cos(x) using differentiation.

   Hint

   Use the properties of exponents and derivatives.

6. 6.

   Using sin(x) = (e^(ix) - e^(-ix)) / (2i), find the integral of sin^2(x) from 0 to π.

   Hint

   Utilize the identity for sine to expand the expression before integrating.