If a line makes angle $\pi/3$ and $\pi/4$ with the positive directions of x-axis and y-axis respectively, then the acute angle made by the line with positive direction of z-axis is

A line in 3D space makes angles α, β, and γ with the positive x, y, and z axes respectively. These angles satisfy:

$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

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Given:

• Angle with x-axis: $\alpha = \frac{\pi}{3}$
• Angle with y-axis: $\beta = \frac{\pi}{4}$
• Angle with z-axis: $\gamma = ?$

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Apply the formula:

$\cos^2\left(\frac{\pi}{3}\right) + \cos^2\left(\frac{\pi}{4}\right) + \cos^2\gamma = 1$

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Calculate the values:

$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

$\cos^2\left(\frac{\pi}{3}\right) = \frac{1}{4}$

$\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

$\cos^2\left(\frac{\pi}{4}\right) = \frac{1}{2}$

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Substitute into the formula:

$\frac{1}{4} + \frac{1}{2} + \cos^2\gamma = 1$

$\frac{1}{4} + \frac{2}{4} + \cos^2\gamma = 1$

$\frac{3}{4} + \cos^2\gamma = 1$

$\cos^2\gamma = 1 - \frac{3}{4}$

$\cos^2\gamma = \frac{1}{4}$

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$\cos\gamma = \pm\frac{1}{2}$

Since the question asks for the acute angle, take the positive value:

$\cos\gamma = \frac{1}{2}$

$\gamma = \frac{\pi}{3}$

Therefore, the acute angle made by the line with the positive direction of the z-axis is $\frac{\pi}{3}$.