A cusp is the moment one sign changes to the next. The cusp between Sagittarius and Capricorn happens at a precise moment, which varies from year to year. If you were born before it you would be Sagittarius, and after it you would be Capricorn.
The moment of the Sagittarius Capricorn cusp is not related to calendar dates. It's not as simple as saying December 21st is always Sagittarius and December 22nd is always Capricorn. One year the cusp may be at 11pm at night on December 21st, and the next year it may be early in the morning on December 22nd etc.
To find out for sure which sign you are you will need to know when exactly you were born. Specifically you will need to know:
With that information an astrologer can calculate which side of the Sagittarius Capricorn cusp you were born on, and therefore which sign you are. For example, this calculation is always performed as part of my natal readings (for one person) and relationship readings (for couples).
If you don't know your birth time then things get a little more complicated, but it is often possible to figure it out "backwards" from major life events. That's a technique called birth time rectification. I won't confuse things here by going into it, other than to say it exists for that scenario.
Another way to establish which sign you are is simply to look at the list of Sagittarius traits compared to Capricorn traits. Sagittarius and Capricorn have very different characteristics so it's often fairly easy to see which one you are.
In some cases however people relate almost equally to both signs. In that situation it's going to be because your sun sign is one of them, while you have another planet (most often mercury) in the other. For example you may have your sun in Sagittarius and your mercury in Capricorn.
In that situation you probably will need an astrologer to tell you which is which, and explain how they interact. There are a large number of permutations so trying to cover them all here would be very confusing, but I automatically include your specific planets and interactions in all my natal readings (for one person) and relationship readings (for couples) mentioned above.
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