What do the sine, cosine and tangent functions all do? [closed]

Question

I know how to find the specific sides given by using $\sin x, \cos x,$ and $\tan x$, but what do these functions actually do, like when you say $\sin 43$, or $\cos 76$, or $\tan 29$?

Answer 1

We only know sin, cos, for some famous angles like 90,60,30,45 other than that we don't know but it is sometimes useful to use trig. identities to find values of other angles, for example we know that $$sin(x-y)=sinx(cosy)-(cosx)siny$$ this can be used to find $sin 15$ for example since $15=45-30$ and we know sin and cos for 45 and 30. other than this the only possible thing to do is approximation we know that $$sinx=\displaystyle \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}+...$$ since this sum is infinite we can choose some cut point lets say n=3 (the larger the cut point the more accurate your approximation) so we have $$sinx\approx x-\frac{x^3}{3!}+\frac{x^5}{5!}$$ so for example 43 in radians is $\frac{43\pi}{180}$ so let $x=\frac{43\pi}{180}$ and you'll get$$sin\frac{43\pi}{180}\approx 0.682$$ and if you used a calculater the first five digits are 0.68199, which is very close, and remember if we chose n is be larger say 10 the approximation will be better.

Remark: we also know some identities for $cosx$ and once we have $sinx$ and $cosx$ we can easily find $tanx$