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/4)

검증된 단계별 안내

Recognize that the expression \(e^{\frac{\pi i}{4}}\) can be rewritten using Euler's formula: \(e^{i\theta} = \cos \theta + i \sin \theta\). Here, \(\theta = \frac{\pi}{4}\).

Substitute \(\theta = \frac{\pi}{4}\) into Euler's formula to express the complex number as \(\cos \left( \frac{\pi}{4} \right) + i \sin \left( \frac{\pi}{4} \right)\).

Calculate the values of \(\cos \left( \frac{\pi}{4} \right)\) and \(\sin \left( \frac{\pi}{4} \right)\), which are both \(\frac{\sqrt{2}}{2}\), to find the rectangular form of the complex number.

Interpret the complex number \(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\) as a point in the complex plane with coordinates \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).

Plot this point on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part, to visualize \(e^{\frac{\pi i}{4}}\).

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주요 개념

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Euler's Formula

Euler's formula states that for any real number θ, e^(iθ) = cos θ + i sin θ. This fundamental relationship connects complex exponentials with trigonometric functions, allowing complex numbers to be represented in polar form.

Complex Numbers in Polar Form

A complex number can be expressed as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). Using Euler's formula, this is written as re^(iθ), which simplifies plotting and multiplication of complex numbers.

Plotting Complex Numbers on the Complex Plane

Complex numbers are plotted on a plane with the real part on the x-axis and the imaginary part on the y-axis. The angle θ from the positive real axis and the magnitude r determine the point's position, making polar form useful for visualization.