Practice Complex Exponential Function (5) - Complex Exponential Function

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6 sample questions

Question 1.
Show that e^(iπ) + 1 = 0 using Euler’s formula.
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Think about the values of cosine and sine at π.
Question 2.
Express cos(3x) using exponential functions.
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Use Euler's formula.
3Question 3.
What does Euler's formula state?
- e^(ix) = cos(x)
- e^(ix) = cos(x) + i*sin(x)
- e^(ix) = sin(x) + i*cos(x)
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Remember, it connects e^(ix) with both cosine and sine.
4Question 4.
True or False: The modulus of e^(x + iy) is |e^x|.
- True
- False
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Consider the definition of modulus for complex numbers.
Question 5.
Prove that e^(iθ) has a modulus of 1 for any real θ. What does this imply about its representation on the complex plane?
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Check the properties of sine and cosine for their squared sums.
Question 6.
Use the properties of complex exponentials to solve for solutions to the equation d²y/dt² + 16y = 0, discussing its relationship to oscillatory motion.
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Think about the connections between roots and oscillation frequency.

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