Math Problem Statement
Solution
To write the complex number in polar form with argument between and , follow these steps:
Step 1: Calculate the modulus
The modulus is given by: where and .
Step 2: Calculate the argument
The argument is given by:
The arctangent of in the coordinate system places the angle in the fourth quadrant. Thus: \theta = \frac{7\pi}{4} \, \text{(since \tan^{-1}(-1)-\frac{\pi}{4}[0, 2\pi])}
Step 3: Write the polar form
The polar form of a complex number is:
Substituting the values and :
Alternatively, using Euler's formula, the polar form can also be expressed as:
Let me know if you'd like any additional clarifications!
Related Questions:
- How can the polar form of a complex number be converted back to rectangular form?
- Why is the argument adjusted to the range ?
- What is the significance of Euler's formula in the context of polar coordinates?
- How do the quadrant positions of a complex number affect the sign of the argument?
- What are the practical applications of representing complex numbers in polar form?
Tip:
Always remember that in polar form, the angle depends on the quadrant of the complex number. Use the signs of and to correctly determine its placement.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Polar Coordinates
Trigonometry
Formulas
r = √(x² + y²)
θ = tan⁻¹(y / x)
z = r(cosθ + i sinθ)
z = re^(iθ)
Theorems
Euler's Formula
Suitable Grade Level
Grades 11-12