Represent the complex number graphically, and find the trigonometric form of the number. 1+3 i

Name: Problem 42, Chapter 7: Algebra and Trigonometry Real Mathematics, Real People
Uploaded: 2020-08-20T23:58:52-08:00
Duration: 2 min 19 s
Channel: Ankit Gupta
Description: Represent the complex number graphically, and find the trigonometric form of the number. 1+3 i

step 1 It is given that the complex number z is equal to be 1 plus 3 aita. We need to plot the complex

Represent the complex number graphically, and find the...

Key Concepts

Trigonometric (Polar) Form

The trigonometric or polar form of a complex number expresses the number using its modulus and argument. It is typically written as r(cos? + i sin?), where r is the modulus and ? is the argument. This form is particularly useful for simplifying multiplication, division, and exponentiation of complex numbers, as well as for analyzing their geometric properties.

Argument

The argument of a complex number is the angle measured from the positive real axis to the line segment connecting the origin and the point representing the number. Often measured in radians, this angle describes the direction of the complex number in the complex plane. Understanding the argument is key to expressing the complex number in trigonometric form, as it defines the orientation of the number.

Modulus

The modulus of a complex number is the distance from the origin to the point representing the number on the complex plane. It is calculated as the square root of the sum of the squares of the real and imaginary parts. The modulus provides a measure of the size or magnitude of the complex number, which is essential in converting a complex number from its rectangular form to its polar (trigonometric) form.

Complex Plane

The complex plane is a two-dimensional space where each complex number is represented as a point. The horizontal axis (real axis) corresponds to the real part of the number, while the vertical axis (imaginary axis) corresponds to the imaginary part. This graphical representation is crucial for visualizing operations on complex numbers, such as addition, subtraction, and understanding magnitudes and angles.

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