Sort through all of the gobbledegook and verbiage of economists, and the thing that finally shakes out is that the value of a coin, and the product it 'buys', is entirely arbitrary. It is always, in the end, determined by the buyer and the seller at the time of the transaction. Sales are, after all, nothing but a barter system. Sellers try to get as much for their wares as they can, and buyers try to pay as little as possible. Each transaction 'value' is determined at the time of the transaction.
However, over many, many transactions, a 'pattern' or 'expectation' develops. The seller expects to get a certain compensation, and the buyer expects to pay a certain compensation, for a certain amount of product. If the buyer does not get the expected product at the expected compensation, the buyer either goes without, tries to find another seller, or raises their expectation. Likewise, if the seller does not get the expected compensation, the seller either does not sell the product, lowers their expectation, or tries to find another buyer.
Profit and cost usually have no bearing on the compensation offered and accepted. Profit is an artifact of the bartering process. If the compensation paid is higher than the costs, then there is a profit. If the compensation paid is lower than the cost to produce, there is a loss. But the compensation paid is ALWAYS a result of the barter process, not of costs. Every seller tries to get the maximum compensation, irrespective of the cost to produce the product. Selling at a loss is not always a bad thing, as it generates new revenue that can be used to produce a product that is more profitable.
What differs from economy to economy is how 'free' the buyer and seller are to negotiate. In some economies, the compensation is arbitrarily decided by some governing body. The pre-determined compensation can be valued according to many, many different criteria, but usually it comes down to a value determined by popularity. That is, does the person or government institution want to be more popular with the buyers, or the sellers? And that decision is based on which of the two has the power to keep the person or government in control. Inevitably, however, there will be buyers and sellers who negotiate 'behind the scenes'. This is known as the 'black market'.
The value of a 'coin', or currency, is determined by how much 'stuff' it can buy. A currency that can buy a lot of stuff, has a high value. A currency that can not buy much stuff, has a low value.
Usually, in a modern economy, 'coinage' is broken down into 'denominations' (pennies, quarters, dollars). These divisions are entirely arbitrary, but they usually involve some sort of equivalent value proportionate to the breakdown. That is, one hundred pennies always makes a dollar. But this is not absolute. 'Making change' is still a barter system. Sometimes, a buyer will pay a dollar for three quarters, if the buyer needs a quarter badly enough.