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

Verified step by step guidance

1. Recognize that the expression \(-e^{-\pi i}\) involves Euler's formula, which states that \(e^{i\theta} = \cos \theta + i \sin \theta\).
2. Identify the angle \(\theta\) in the expression \(e^{-\pi i}\). Here, \(\theta = -\pi\).
3. Apply Euler's formula: \(e^{-\pi i} = \cos(-\pi) + i \sin(-\pi)\).
4. Calculate \(\cos(-\pi)\) and \(\sin(-\pi)\) using the unit circle: \(\cos(-\pi) = -1\) and \(\sin(-\pi) = 0\).
5. Substitute these values back into the expression: \(e^{-\pi i} = -1 + 0i\), then multiply by \(-1\) to get \(-e^{-\pi i} = 1 + 0i\), which is the complex number to plot.

Key Concepts

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 for the representation of complex numbers in polar form. Understanding this formula is crucial for converting between exponential and trigonometric forms of complex numbers.

Complex Numbers

A complex number is expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit defined as the square root of -1. The real part (a) represents the horizontal axis, while the imaginary part (b) represents the vertical axis in the complex plane. This representation is essential for visualizing and manipulating complex numbers.

Polar Coordinates

Polar coordinates provide a way to represent complex numbers using a magnitude (r) and an angle (θ). In this system, a complex number can be expressed as r(cos(θ) + i sin(θ)), which corresponds to its position in the complex plane. This approach simplifies multiplication and division of complex numbers and is particularly useful when working with Euler's formula.