destiny
The main guaranteed source of Three of Coins is to purchase them from Xûr on the weekend, where you get 5 Three of Coins for 7 Strange Coins.
I believe there are other ways, could be random, could be guaranteed rewards from quests.
What are the different ways to earn Three of Coins, what sources do exist?
Golden Chests just contain loot, yes. It's also worth noting that they scale on the area, not your level. So it's best to find them asap. You also are given an item for finding all the golden chests on a single planet. I know that you get a Sparrow from Earth and an Emblem from the Moon.
There's a few on each of the planets in-game at the moment. There are 5 on the moon
1 - Found during the The World’s Grave mission inside the cave full of cocoons.
2 - Head to the mission marker located in the large yellow building. Proceed until you find the green gas pit. Look nearby for a rock formation and orange lights.
3 - Another in The World’s Grave. When defending your Ghost search a nearby room with a little pool of water
4 - Inside the Temple of Crota during the Chamber of Night mission.
5 - The Summoning Pits Strike mission. After the final boss fight walk to the edge of the room and drop down onto a small platform
I'm affiliated with Gamers Heroes and we have a guide on the topic should you need more direction: Destiny Gold Chests Moon
You can use 10 legendary marks to purchase 20 of any planetary material from the Vanguard Quartermaster. See below to locate the quartermaster.
Faction rep (per each of the 3 factions: Dead Orbit, Future War Cult, New Monarchy) can be gained by speaking to the respective faction rep in the tower, as seen on the below map. The exchange rates are as follows, gaining rep for the faction rep you exchange with:
5 Motes -> 100 rep
25 Weapon Parts -> 50 rep
25 Old Armor Materials✝ -> 50 rep
25 Unified Armor Materials -> 50 rep
Special Ammo Synth -> 25 rep
Heavy Ammo Synth -> 25 rep
✝ Old Armor Materials includes Plasteel Plating, Sapphire Wire, and Hadronic Essence.
EDIT: The below exchanges are no longer valid as of "The Taken King", September 2015.
You can now go to the Vanguard/Crucible Quartermaster and exchange 10 marks for:
20 Spinmetal
20 Helium Filaments
20 Spirit Bloom
20 Relic Iron
The Crucible Quartermaster is in the middle area of the tower before the class Vanguards (opposite Lord Shaxx). On the map this is in the blue area near the icon that says Crucible.
The Vanguard Quartermaster is in the area near the shipwright. On the map this is in the red area and marked Quartermaster.
EDIT: The below is no longer true as a of a patch in late 2014.
Each time you exchange 50 of each material to either the Crucible or Vanguard Quartermaster, you get:
+5 Crucible/Vanguard Marks (as long as you're below the weekly limit)
+25 Crucible/Vanguard Reputation (these change if you have a faction class armor equipped)
+250 Glimmer
Note that you are also capped at 200 total marks for each faction, and 100 per week for each faction.
Best Answer
Three of Coins can be acquired from the nightfall rewards. The patch from 20 october 2015 shows that the amount of coins that can drop has been increased, so naturally they drop there. They can't drop anywhere else from what has been documented and as far i know it doesn't drop in the Kings Fall raid.
As a bit of side info, here is a small guide on Three of Coins