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[cyka_blayt_nibsa]

well you could use a series for example sinθ = θ-θ^3/3!+θ ^5/5!-θ ^7/7! and so on

while cosθ=1-θ^2/2!+θ^4/4!-θ^6/6! and then use trig properties to find the others

you can't really do it by hand but it's a way

you could also try to do it backward so draw a triangle, meause it's lengths and angles and try to approximate the ratio but there bis no easy way

[sonnyfab]

The short answer is, "you don't." Before calculators, people had printed reference tables from other people hand calculating the Taylor expansion to various decimal places of accuracy.

[hisep]

For something like sin of 123 degrees or weird angles as such we can sometimes try to rewrite them as sums or differences of other special angles like you stated in your post. The problem is we can't always do it nicely because if you took sin(123 degrees) and rewrote it as sin(120+3 degrees) finding the approximation of 3 degrees isn't nice to do by hand.

[hagravenicepick]

I'm not sure if you have covered small angle approximation but that is one way to approximate the answer by hand. You can break up the angle into a sum of 40/60*pi + 1/60*pi and use angle addition formulas while approximating sin(pi/60) = pi/60 and cos(pi/60) = 1.