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You can always find your latitude and longitude by getting a 3-star celestial fix; as long as you have a sextant and an accurate timepiece.

If you are looking for a one-star observation to determine your latitude, then you would need to be able to see Polaris, the north star. Polaris cannot be observed anywhere south of about 3 degrees north latitude. (Polaris is not EXACTLY above the North Pole, so in order to get an accurate reading of your longitude, you would still need a nautical almanac and a watch.)

But with a watch, a sextant and a nautical almanac, you could do a "local apparent noon" observation of the Sun, and determine both your latitude and longitude quite precisely from one sighting.

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