There are a few ways. First, there are a multitude of trigonometric tables which list the sines and cosines of a variety of values. if you now one trigonometric value of a number, you can find all the others by hand, and you can also use a Taylor series approximation to find a fairly accurate value. (In fact, many calculators use Taylor series to find trigonometric values.)
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subtract 90 from it and find the trig ratio of that and it will be equal to the trig ratio that is over 90 degrees
Nowadays you cannot do it without a calculator.
You cannot.
Use a ruler
To find an angle without using a calculator, you will need to use a trig identity. Determine which angle it is and use its corresponding trig identity.