Math Problem Statement
Solution
The problem requires representing complex numbers in their trigonometric form. Let me break it down step by step for each case.
The general trigonometric form of a complex number is: where:
- is the modulus of the complex number,
- is the argument (angle) of the complex number.
a)
Here, is purely real and negative. Its components are:
- Real part ,
- Imaginary part .
-
Find the modulus :
-
Find the argument : Since lies on the negative real axis, the argument is:
-
Write the trigonometric form:
b)
Here, the components of are:
- Real part ,
- Imaginary part .
-
Find the modulus :
-
Find the argument : The complex number is in the fourth quadrant (real positive, imaginary negative). The angle can be determined as: \theta = -\frac{\pi}{6} \, \text{(radians)} \quad \text{(or equivalently \frac{11\pi}{6})}.
-
Write the trigonometric form:
Final Answer:
a)
b)
Would you like a more detailed explanation for any of the steps?
Follow-Up Questions:
- How do you convert rectangular form to polar form for complex numbers?
- What is the significance of the modulus and argument of a complex number?
- Can you generalize the trigonometric form for real and imaginary axes?
- How do you find the argument for complex numbers in different quadrants?
- How is Euler's formula related to the trigonometric form of complex numbers?
Tip: The modulus and argument are critical when working with complex numbers in polar form, especially for multiplication and division operations.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Trigonometric Form
Modulus and Argument
Polar Representation
Formulas
Trigonometric Form: z = r (cos θ + i sin θ)
Modulus: r = √(x² + y²)
Argument: θ = arctan(y / x)
Theorems
Polar form of complex numbers
Properties of angles in the complex plane
Suitable Grade Level
Grades 10-12