In my book, Bob takes off on a spaceship with a light clock so that the direction of motion of his spaceship is perpendicular to the direction of motion of light in the light clock.

As a result, Alice (on Earth) sees the distance traveled by light to be $\sqrt{L^2+(vt)^2}$, (given by Pythagoras theorem) which leads to the usual time dilation formula.

But if Bob's spaceship doesn't travel perpendicular to the light, then the distance observed by Alice would be given by law of cosines. This will lead to a different formula for time dilation

Why is the second formula not valid?

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    $\begingroup$ There are existing questions here on this topic, I'll try to find one. Briefly, if the light path in the clock isn't perpendicular to the ship motion, then Alice will need to take length contraction into account. $\endgroup$ – PM 2Ring Jul 28 '19 at 6:13
  • $\begingroup$ Also see physics.stackexchange.com/q/276574/123208 and the links there. $\endgroup$ – PM 2Ring Jul 28 '19 at 6:18