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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Show that e^(iπ) + 1 = 0 using Euler’s formula.
💡 Hint: Think about the values of cosine and sine at π.
Question 2
Easy
Express cos(3x) using exponential functions.
💡 Hint: Use Euler's formula.
Practice 3 more questions and get performance evaluation
Interactive Quizzes
Engage in quick quizzes to reinforce what you've learned and check your comprehension.
Question 1
What does Euler's formula state?
- e^(ix) = cos(x)
- e^(ix) = cos(x) + i*sin(x)
- e^(ix) = sin(x) + i*cos(x)
💡 Hint: Remember, it connects e^(ix) with both cosine and sine.
Question 2
True or False: The modulus of e^(x + iy) is |e^x|.
- True
- False
💡 Hint: Consider the definition of modulus for complex numbers.
Solve and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Prove that e^(iθ) has a modulus of 1 for any real θ. What does this imply about its representation on the complex plane?
💡 Hint: Check the properties of sine and cosine for their squared sums.
Question 2
Use the properties of complex exponentials to solve for solutions to the equation d²y/dt² + 16y = 0, discussing its relationship to oscillatory motion.
💡 Hint: Think about the connections between roots and oscillation frequency.
Challenge and get performance evaluation