Step 1: Understanding the Question:
The question asks for the argument (angle) of a complex number given in a fractional form. To find the argument, we first need to simplify the complex number into the standard form \(z = a + bi\).
Step 2: Key Formula or Approach:
To simplify a fraction of complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator.
If \(z = a + bi\), the argument is given by \(\text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right)\), considering the quadrant in which the point (a, b) lies.
Step 3: Detailed Explanation:
Given complex number \( z = \frac{13-5i}{4-9i} \).
The conjugate of the denominator (4 - 9i) is (4 + 9i).
Multiply the numerator and denominator by (4 + 9i):
\[
z = \frac{(13-5i)(4+9i)}{(4-9i)(4+9i)}
\]
Calculate the numerator:
\[
(13-5i)(4+9i) = 13(4) + 13(9i) - 5i(4) - 5i(9i)
\]
\[
= 52 + 117i - 20i - 45i^2
\]
Since \(i^2 = -1\):
\[
= 52 + 97i - 45(-1) = 52 + 97i + 45 = 97 + 97i
\]
Calculate the denominator:
\[
(4-9i)(4+9i) = 4^2 - (9i)^2 = 16 - (81i^2)
\]
\[
= 16 - 81(-1) = 16 + 81 = 97
\]
Now, substitute the numerator and denominator back into the expression for z:
\[
z = \frac{97 + 97i}{97} = \frac{97(1+i)}{97} = 1 + i
\]
The complex number is \(z = 1 + 1i\). This is in the form \(a + bi\), with \(a = 1\) and \(b = 1\).
Since both \(a\) and \(b\) are positive, the complex number lies in the first quadrant.
The argument is:
\[
\text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4}
\]
Step 4: Final Answer:
The argument of the complex number z is \( \frac{\pi}{4} \).