In calculus, it can be shown that e^(iθ) = cos θ + i sin θ. In Exercises 87–90, use this result to plot each complex number. -e^-πi

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Recall Euler's formula: \(e^{i\theta} = \cos \theta + i \sin \theta\). This expresses a complex number on the unit circle in terms of an angle \(\theta\).

Identify the angle \(\theta\) in the expression \(-e^{-\pi i}\). Notice that \(e^{-\pi i} = \cos(-\pi) + i \sin(-\pi)\) by Euler's formula.

Calculate \(e^{-\pi i}\) using the cosine and sine values: \(\cos(-\pi) = \cos \pi\) and \(\sin(-\pi) = -\sin \pi\). Substitute these values to get the complex number.

Multiply the result by \(-1\) to account for the leading negative sign in \(-e^{-\pi i}\). This will reflect the point across the origin in the complex plane.

Plot the resulting complex number on the complex plane, where the real part corresponds to the x-axis and the imaginary part corresponds to the y-axis.

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주요 개념

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Euler's Formula

Euler's formula states that e^(iθ) = cos θ + i sin θ, linking complex exponentials to trigonometric functions. It allows expressing complex numbers in polar form, facilitating their interpretation as points or vectors in the complex plane.

Complex Number Representation in the Complex Plane

Complex numbers can be represented as points or vectors in the complex plane, where the x-axis is the real part and the y-axis is the imaginary part. Using Euler's formula, a complex number can be plotted by interpreting its magnitude and angle (argument).

Properties of Exponents and Negative Signs in Complex Numbers

Understanding how to handle negative signs and exponents in complex numbers is essential. For example, -e^(-πi) involves both a negative sign and a complex exponential with a negative angle, affecting the position and direction of the point in the complex plane.