Is there a method to find the values given on the unit circle without drawing it out?

Question

Is it possible to solve the sin/cos/tan of an angle and/or their inverses without a calculator, using the Taylor expansion, or looking at a unit circle? Recently I've just made myself memorize the unit circle and solved from there, but I know that can only take me so far.

No, not really. However one can get approximations to a desired accuracy using Taylor expansions (I know, you wanted to avoid this), and using the first order approximation will give you fairly good results just by knowing the value of a trig function at a nearby point. $\sin(x) \approx \sin(x_0) + \cos(x_0) (\frac{x-x_0}{2\pi})$
Where $x_0$ is a point nearby $x$ which you know the value of sine and cosine at. (note that you'll want to mod $x$ by $2 \pi$) — infinitylord, Dec 16 '17 at 00:53
Please, if you are ok, you can accept the answer and set it as solved. Thanks! — user, Jan 28 '18 at 23:33

score 0 · Answer 1 · answered Dec 16 '17 at 00:53

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I think you should memorize the values (or use the unit circle) for a few common multiples of $\pi$ (denominators $1$, $2$, $3$, $4$ and $6$). After that you need a calculator or tables or Taylor series or other numerical methods.

answered Dec 16 '17 at 00:53

Ethan Bolker

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score 0 · Accepted Answer · answered Dec 16 '17 at 01:06

Note that it suffice to memorize the important values in the first quadrant/octant and then obtain the others by symmetry and basic trigonometric identities.

Is there a method to find the values given on the unit circle without drawing it out?

2 Answers2