here, here, and here (just don't believe all that magnet nonsense). I'm pretty good at it. This post is about when none of that works.
Step 1 - If I'm not sure I've triggered a light signal, I'll often roll the bike back & forth, to give the induction sensor another chance. If it's a camera, I'll wave my arms & might waddle the bike over to the RH side of the left lane if it looks like the camera alignment is skewed. Sometimes a car will come along behind and do the honors, obviating the need for further action.
Step 2 - I've been known to lay the bike down, run over to hit the pedestrian crossing signal & then run back to the bike. This is not a recommended approach because you're an easy target for a right hook when doing this, not to mention falling if you've fallen victim to the "road bike clipless" propaganda machine - and think of how you'd feel if the light changed right as you were running up to the button. It is dramatic, however. You'd also be really miffed if someone came along and ran over your bike while you were engaging in such folly. Also not recommended - turning the bike 90 degrees to help the camera "see" it, though that might be better than the "Lemans" button sprint.
Step 3 - The free right turn. If I can still get where I want to go, I'll make a free right turn and then thread my way back to where I'm going. This works on my commute since I'm generally going from SE to NW. Step 4 complaints apply for step 3. Step 3 is also handy for a few lights with very long cycles.
Step 4 - I'll wait through two light cycles and then I go when it's safe. That will be followed up by a complaint to the city in question about their improperly operating signal. While I often call the police non emergency number, the correct place to call is probably the Department of Public Works. Remember, we need to help our motorcycle brethren out on this. Remember, if you get hit while crossing, the headline will probably read "cyclist killed crossing XXX" rather than "inoperative traffic signal leads to cyclist fatality."
If I were KING OF THE UNIVERSE and could specify only two bicycle/motorcycle facility items (other than bike racks everywhere), I'd wish for "sweet spot" markings on the pavement (who says paint is bad?) and for some sort of visual indicator (maybe a small green LED or something below the camera) so a cyclist or biker could know the signal sequence had been triggered. Now that would be "bicycle friendly."