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3

Ahh, Richard Fitzpatrick. Great guy. Ok, If you start with the second set of expressions, use the appropriate double-angle-formula and then assume the "angle" $2\Omega \sin\lambda t$ is small (note that the $t$ is not within the sin function!), you get the first expressions, e.g. $$\cos(\theta+\phi) = \cos\theta\cos\phi - \sin\theta\sin\phi,$$ and then ...

3

Note that Fitzpatrick states towards the beginning, The following solution method exploits the fact that the Coriolis force is much smaller in magnitude that the force of gravity: hence, $\Omega$ can be treated as a small parameter Generally, when statements like that are made, powers (greater than 1) of the term in question are considered to be zero: ...

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Indirectly, yes. It is called a gyro compass. When you constrain a gyroscope to spin about an axis in the horizontal plane it will experience a Coriolis like force unless it is pointing due North-South - if it isn't it will slowly align, then stay there, so you can use this as a compass - in fact it is widely used as it doesn't suffer from magnetic ...

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