The U.S. never minted a half dollar gold coin.
If your coin is the size of a standard half dollar, it's been plated for use in jewelry or as a keepsake. Unfortunately, that undoes any extra numismatic value it may have had.
If your coin is smaller or does not look like the coin shown in this picture, you could have either what's called a territorial gold coin, privately issued to address coin shortages in the West, or more likely you have a replica. Lots of gift shops in the old mining areas sell replica coins. Replicas made after 1973 are supposed to have the word COPY on them to make sure no one thinks they're real, but any that are older can easily be confused.
There are multiple coins that might bear the III so you need to first identify it before we can assign a value to it. A slightly unlikely coin is the US three-cent piece silver, on the front it has a star with a shield in it. The other possible US coin is the US three-cent piece nickel, this coin has a personification of Liberty as a woman on the front of it. Both coins were minted with the date of 1868 with the three-cent silver being quite rare and the three cent nickel being worth around $3-7 in circulated condition.
This is likely a privately minted medal. If it's silver, it's value will be based on that silver it contains -- currently at about $12 per ounce. If it's not silver, then it's worth about a dollar.
Probably
$200
One dollar. In the future - DECADES into the future - such a coin may have a higher value to a coin collector based on scarcity and condition.
This coin came out in 2012. It sells for a premium from the mint. They have a face value of one-dollar but cost about $1.50 and are only available in sets, so if you can find a buyer they should be worth that much. Proofs are a little more.
$15.00 to $30.00 in circulated condition
Assuming you mean a Morgan silver dollar and not a gold one dollar coin, if the coin is circulated and has no mintmarks, retail value is $26.00-$32.00 depending on how much wear the coin has. The 1883 Morgan is a common date.
The 1883 Morgan dollar is common with retail values for circulated coins at $18.00-$30.00 and uncirculated coins at $40.00-$60.00
1883 Morgan dollars are very common, uncirculated coins with no mintmark sell for $50.00-$65.00. The actual value depends on the grade of the coin.
One dollar
A circulated coin has a value of between 32.00-39.00 dollars today.
Assuming the coin is circulated and has no mintmark, the 1883 Morgan dollar is a very common coin with retail values of $31.00-$39.00 depending on the grade of the coin.
1848. Silver. Twenty. Dollar. Coin
Assuming the coin is circulated and has no mintmark, the 1883 Morgan dollar is a very common coin with retail values of $32.00-$39.00 depending on the grade of the coin.
The 1883 Morgan is a very common coin. It is not considered rare or even scarce.
With a date of 1883 the coin is not a silver eagle it's a Morgan dollar and is very common in mint state condition. Retail values are $28.00-$50.00 depending on grade.
There is no such thing as a 1960 silver dollar coin.