Finding sin and cos without using calculator

[Emmanuel_Euler]

Hi everyone.
i think this is my last thread on PF!
because i am too busy,anyway,
4 months ago i posted a thread named it finding cube roots without using calculator and now i want to know if there is a way or method to find sin and cos without using calculator.
And thanks to all who helped me in all of my questions.

 

[axmls]

You can get as close as you want by using a Taylor series.

\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ ...

\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ ...

This works well for x near 0. Just cut it off at however many terms you want depending on how accurate you want it. You could use a more general form for the functions starting at any x = a, but given their periodicity, it seems the simplified would work fine. The only issue then is adding up some fractions.

 

[SteamKing]

You can get as close as you want by using a Taylor series.

\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ ...

\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ ...

This works well for x near 0. Just cut it off at however many terms you want depending on how accurate you want it. You could use a more general form for the functions starting at any x = a, but given their periodicity, it seems the simplified would work fine. The only issue then is adding up some fractions.

Maybe you can use a Taylor series to evaluate a trig function without pressing the sin x or cos x keys on a calculator, but I think actually evaluating the Taylor series without using the other calculator functions will get a bit tedious, especially as you add more terms to the evaluation.

 

[Emmanuel_Euler]

Maybe you can use a Taylor series to evaluate a trig function without pressing the sin x or cos x keys on a calculator, but I think actually evaluating the Taylor series without using the other calculator functions will get a bit tedious, especially as you add more terms to the evaluation.

should i have a Calculator function to evaluate taylor series??

 

[SteamKing]

should i have a Calculator function to evaluate taylor series??

No. And your calculator wouldn't use Taylor series to calculate the value of sin x or cos x, either. The Taylor series is slow to converge, and calculators with built-in trig functions use different methods to calculate their values.

https://en.wikipedia.org/wiki/CORDIC

 

[thegirl]

What came to my mind when you said that was draw out a triangle with the angle you want to measure and then divide the hypothenuse by the adjacent for cos and the opposite for sin??

 

[VKnopp]

This is similar to what SteamKing wrote. The $\cos z$ and $\sin z$ functions respectively are just real and imaginary parts of $e^{iz}$. We can calculate the sine and cosine functions (in radians) by calculating the real and imaginary parts of the series,

\sum_{n=0}^{\infty} {\frac{(-1)^{\frac{n}{2}} z^n}{n!}}