Math Problem Statement
Represent 7zi into polar form and find the argument of z and the principal value of the argument of 𝑧.
Solution
To represent in polar form and find the argument and the principal value of the argument, we’ll proceed step-by-step.
Step 1: Convert to Polar Form
-
Identify in Cartesian coordinates: Here, , which means it has no real part, so: Therefore, (real part) and (imaginary part).
-
Find the modulus : The modulus is calculated as:
-
Calculate the argument : Since lies on the positive imaginary axis, the argument is:
Thus, the polar form of is: or, in exponential form:
Step 2: Find the Argument of and the Principal Value
-
Argument of : The argument of , , is .
-
Principal Value of the Argument: The principal value of the argument, , is the unique value of the argument within . Here, the principal argument is also:
Summary:
- Polar Form: or
- Argument :
- Principal Argument :
Would you like further details on any part of this process?
Here are some related questions:
- How do you convert complex numbers into polar form?
- What is the difference between argument and principal argument?
- Why is the modulus of a complex number always positive?
- How does the argument of a complex number change with quadrants?
- How do you express complex numbers on the Argand plane?
Tip: For complex numbers lying on the imaginary axis, the argument is always , depending on the sign of the imaginary part.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Complex Numbers
Polar Coordinates
Trigonometry
Formulas
Polar form of a complex number: \( z = r(\cos \theta + i \sin \theta) \) or \( z = r e^{i\theta} \)
Modulus of a complex number: \( |z| = \sqrt{x^2 + y^2} \)
Argument of a complex number: \( \theta = \tan^{-1}(\frac{y}{x}) \)
Theorems
Principal Argument of a Complex Number
Suitable Grade Level
Grades 11-12