e to the i pi equals what value?

e to the i pi equals what value?

• A. i
• B. -1
• C. 1
• D. -0

Answer: (B)

The key idea is how complex exponentials relate to trigonometry. Euler’s formula tells us e^{iθ} = cos θ + i sin θ. So the value depends on the angle θ on the unit circle. For θ = π, cosine is -1 and sine is 0, giving e^{iπ} = -1 + 0i = -1. The imaginary part vanishes, and the point lies on the negative real axis. This matches the well-known relation e^{iπ} + 1 = 0 from Euler’s identity. So the value is -1.

The key idea is how complex exponentials relate to trigonometry. Euler’s formula tells us e^{iθ} = cos θ + i sin θ. So the value depends on the angle θ on the unit circle. For θ = π, cosine is -1 and sine is 0, giving e^{iπ} = -1 + 0i = -1. The imaginary part vanishes, and the point lies on the negative real axis. This matches the well-known relation e^{iπ} + 1 = 0 from Euler’s identity. So the value is -1.