Using the Blue Trinket

As ADC with global ults (or honestly anyone with global or semi-global ults), use farsight orb to help aim your ult. As an immobile ADC, basically use it to check bushes, like you would with a ward. It gives you shorter duration, but that's enough for checking bush. It's not useful for maintaining vision of the bush, but ADC isn't useful dead, so...

You will use it primarily to check bushes/baron or dragon pits/ and occasionally so that you can keep attacking someone who is hiding in a bush (whom you were already killing)

Just use it to prevent facechecking bushes.

In LCS or competitive play, most players use it to scout out a bush where a potential team fight can take place. Also as an ADC and you are chasing someone, if they go in a bush use the orb and it will reveal them. Jinx, Draven, Ashe, etc. those with global ultimates, farsight orb is good to check how low the objective is to kill it with a global ultimate.

You use it on anyone with a global skill or far reaching skill. Like Caitlyn's ult or Ashe's. Also to prevent facechecking brushes or just spying on objectives.

Blue trinket is like a safer version of the yellow trinket, mainly for the ADC. Use it on bushes that you're about to walk into.

Safer bush check, spying objectives, and my favorite: spying on the enemy shop keeper. He does some freaky shit while you guys are fighting over Baron.