We are given a complex number of the form:
[tex]w=\frac{2p+12}{13}+\frac{3p-8}{13}i[/tex]
We are also given that arg(w) = π/4. With this information, we can calculate the value of p.
The argument of a complex number is defined as:
[tex]\tan w=\frac{y}{x}[/tex]
Where y and x are the imaginary and real parts (respectively) of the complex number. Applying the formula:
[tex]\tan \frac{\pi}{4}=\frac{\frac{3p-8}{13}}{\frac{2p+12}{13}}[/tex]
Since the tangent of π/4 is 1, the real and the imaginary parts happen to be equal, that is:
[tex]\begin{gathered} \frac{\frac{3p-8}{13}}{\frac{2p+12}{13}}=1 \\ \frac{3p-8}{13}=\frac{2p+12}{13} \end{gathered}[/tex]
Simplifying by 13:
[tex]\begin{gathered} 3p-8=2p+12 \\ \text{Simplifying:} \\ 3p-2p=12+8 \\ \text{Solving:} \\ p=20 \end{gathered}[/tex]
Substituting into the complex number:
w = 3 + 3i
