To solve the problem of finding \( \theta \), given a line that makes angles of \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) with the positive \( x \)- and \( y \)-axes respectively, we start by using the property of the direction cosines. For a line making angles \( \alpha, \beta, \gamma \) with the \( x, y, z \)-axes respectively, the direction cosines \( l, m, \) and \( n \) are given by:
\( l = \cos \alpha \), \( m = \cos \beta \), and \( n = \cos \gamma \).
These satisfy the equation:
\( l^2 + m^2 + n^2 = 1 \).
Substitute the given angles:
\( l = \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \).
\( m = \cos \frac{\pi}{3} = \frac{1}{2} \).
Now, solve for \( n \):
\(\left(-\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + n^2 = 1\).
\(\frac{2}{4} + \frac{1}{4} + n^2 = 1\).
\(\frac{3}{4} + n^2 = 1\).
\(n^2 = 1 - \frac{3}{4} = \frac{1}{4}\).
\(n = \pm \frac{1}{2}\).
\(n = \cos \gamma = \cos \theta = \pm \frac{1}{2}\).
Thus, the values of \( \theta \) are:
\(\theta = \pm \frac{\pi}{3}\).
The correct answer is \( \pm \frac{\pi}{3} \).