In Exercises 37–44, find the product of the complex numbers. Leave answers in polar form. z₁ = cos π/4 + i sin π/4 z₂ = cos π/3 + i sin π/3

검증된 단계별 안내

Recall that complex numbers in the form \(z = \cos \theta + i \sin \theta\) are expressed in polar form, where \(\theta\) is the argument (angle) of the complex number.

To find the product of two complex numbers in polar form, use the formula: if \(z_1 = \cos \alpha + i \sin \alpha\) and \(z_2 = \cos \beta + i \sin \beta\), then their product is \(z_1 z_2 = \cos(\alpha + \beta) + i \sin(\alpha + \beta)\).

Identify the angles for the given complex numbers: \(\alpha = \frac{\pi}{4}\) for \(z_1\) and \(\beta = \frac{\pi}{3}\) for \(z_2\).

Add the angles: calculate \(\alpha + \beta = \frac{\pi}{4} + \frac{\pi}{3}\), which will be the argument of the product.

Write the product in polar form as \(z_1 z_2 = \cos\left(\frac{\pi}{4} + \frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{4} + \frac{\pi}{3}\right)\).

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Complex Numbers in Polar Form

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form highlights the geometric interpretation of complex numbers as points or vectors in the plane, making multiplication and division more intuitive.

Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, multiply their magnitudes and add their angles. Specifically, if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁z₂ = r₁r₂ [cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]. This simplifies the product calculation significantly.

Euler's Formula and Trigonometric Representation

Euler's formula states that e^(iθ) = cos θ + i sin θ, linking exponential and trigonometric forms of complex numbers. This connection helps understand the polar form and the behavior of complex multiplication as rotation and scaling in the complex plane.