In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3 + 4i

검증된 단계별 안내

Identify the complex number given: \(-3 + 4i\). Here, the real part is \(-3\) and the imaginary part is \(4\).

Plot the complex number on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. So, plot the point at coordinates \((-3, 4)\).

Calculate the magnitude (or modulus) \(r\) of the complex number using the formula \(r = \sqrt{(\text{real part})^2 + (\text{imaginary part})^2}\). Substitute the values to get \(r = \sqrt{(-3)^2 + 4^2}\).

Find the argument (or angle) \(\theta\) of the complex number using the formula \(\theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right)\). Substitute the values to get \(\theta = \tan^{-1}\left(\frac{4}{-3}\right)\). Remember to consider the quadrant where the point lies to determine the correct angle.

Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\) or \(r \operatorname{cis} \theta\), where \(r\) is the magnitude and \(\theta\) is the argument you found.

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Complex Numbers and the Complex Plane

A complex number is expressed as a + bi, where a is the real part and b is the imaginary part. It can be represented as a point or vector in the complex plane, with the x-axis as the real axis and the y-axis as the imaginary axis. Plotting involves locating the point (a, b) on this plane.

Polar Form of Complex Numbers

Polar form represents a complex number using its magnitude (distance from origin) and argument (angle with the positive real axis). It is written as r(cos θ + i sin θ) or r∠θ, where r = √(a² + b²) and θ is the angle measured in degrees or radians.

Calculating the Argument (Angle)

The argument θ of a complex number is the angle formed with the positive real axis, found using θ = arctan(b/a). Care must be taken to determine the correct quadrant based on the signs of a and b, ensuring the angle accurately reflects the complex number's position.