Math Problem Statement

argument de sqrt(3)+i

Solution

To find the argument of the complex number z=3+iz = \sqrt{3} + i, we can use the formula for the argument:

arg(z)=tan1(yx)\text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right)

where xx is the real part and yy is the imaginary part of the complex number. Here, x=3x = \sqrt{3} and y=1y = 1.

Calculating the argument:

arg(z)=tan1(13)\text{arg}(z) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)

Since tan(π6)=13\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}, we find:

arg(z)=π6\text{arg}(z) = \frac{\pi}{6}

Thus, the argument of 3+i\sqrt{3} + i is π6\frac{\pi}{6} radians.

Do you want more details or have any questions?

Here are some related questions you might consider:

  1. What is the modulus of the complex number 3+i\sqrt{3} + i?
  2. How do you convert this complex number to polar form?
  3. What is the geometric representation of 3+i\sqrt{3} + i in the complex plane?
  4. How would the argument change if the imaginary part were negative?
  5. Can you explain the significance of the argument in complex analysis?

Tip: Always remember that the argument is the angle formed with the positive x-axis in the complex plane.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Argument of a Complex Number
Trigonometry

Formulas

arg(z) = tan^(-1)(y/x)

Theorems

Inverse Tangent

Suitable Grade Level

Grades 10-12