Trig by hand?

Question

I know this has been asked a lot, but the answers I've been through only go so far ...

How can you do sine, cosine and tangent by hand for any angle? And when I say by "hand". I mean a formula.

Thanks so much!

Answer 1

You would need to use the tedious Taylor expansion of each.

sin x = x - (x^3)/3! + (x^5)/5! - ... ∀x
cos x = 1 - (x^2)/2! + (x^4)/4! - ... ∀x
tan x = x + (x^3)/2 + (2x^5)/15 + ... for |x| < pi/2

I should have noted that x is in radians and ! defines a factorial.

Answer 2

In x y plane , origin O.
Vector OR has magnitude r.
Ɵ is angle between r and x axis.
sin Ɵ = y/r
cos Ɵ = x/r
tan Ɵ = y/x
Leads to (for purposes of allocating signs):-

sin | All
----|----
tan| cos