If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of the z-axis.

Solution

suppose the direction cosines of the line be l,m,and n.
we know that l² + m²+n² = 1
Let the line make angle θ with the positive direction of the z-axis.
α = 90°, β = 60° γ = θ
Thus,
cos² 90 + cos²60 + cos²θ =1
$rightwards double arrow space 0 space plus space open parentheses 1 half close parentheses squared space plus space cos squared straight theta space equals space 1 rightwards double arrow space cos squared space straight theta space equals space 1 minus 1 fourth rightwards double arrow space cos squared space straight theta space equals 3 over 4 rightwards double arrow space cos space straight theta space equals space plus-or-minus fraction numerator square root of 3 over denominator 2 end fraction rightwards double arrow space straight theta space equals space 30 to the power of straight o$

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Inverse Trigonometric Functions

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of the z-axis.

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