Calculating angles for tetrahedral molecular geometry

Let's say I have a molecular like CH3F (i.e. fluoromethane), and I'm able to measure the angle (theta) between the C-H bonds. Provided (theta) what is the angle between the C-F bond and the C-H bonds?

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Use a coordinate system with the C at the origin and the F on the $z$ axis, and one of the H's in the $xz$ plane. Let $\alpha$ be the angle between the C-F and C-H bonds and $\theta$ be the angle between two C-H bonds. The Cartesian components of unit vectors in the directions of the 3 H's are $$(\sin\alpha,0,\cos\alpha),(\sin\alpha\cos 2\pi/3,\sin\alpha\sin 2\pi/3,\cos\alpha),(\sin\alpha\cos 4\pi/3,\sin\alpha\sin 4\pi/3,\cos\alpha).$$ The dot product of any two of these should equal $\cos\theta$: $$\cos\theta=\sin^2\alpha\cos 2\pi/3+\cos^2\alpha=-{1\over 2}\sin^2\alpha+\cos^2\alpha={3\over 2}\cos^2\alpha-{1\over 2}.$$ The solution is $$\cos\alpha=\pm\sqrt{1+2\cos\theta\over 3}.$$ The negative root is the relevant one here, since we know $\alpha$ is obtuse.

If you happen to remember that the angle for methane is $\theta=\cos^{-1}(-1/3)$, you can check that this makes sense: the above formula gives $\alpha=\theta$ in that case, as it should.

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