Nowadays this is done most easily with an advanced graphing calculator.
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trig
You can calculate that on any scientific calculator - like the calculator on Windows (if you change the options, to display as a scientific calculator). Log base 4 of 27 is the same as log 27 / log 4. You can use logarithms in any base to calculate that - just use the same base for both logarithms.
Yes, sine is a trig function, it is opposite over hypotenuse.
You divide log 8 / log 16. Calculate the logarithm in any base, but use the same base for both - for example, ln 8 / ln 16.
Surveying is all trig. Much trig in physics and engineering,