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Finding sin and cos without using calculator

  1. Oct 9, 2015 #1
    Hi everyone.
    i think this is my last thread on PF:frown:!!
    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.
  2. jcsd
  3. Oct 9, 2015 #2
    You can get as close as you want by using a Taylor series.

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

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

    This works well for [itex]x[/itex] near [itex]0[/itex]. 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 [itex]x = a[/itex], but given their periodicity, it seems the simplified would work fine. The only issue then is adding up some fractions.
  4. Oct 10, 2015 #3


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    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.
  5. Oct 10, 2015 #4
    should i have a Calculator function to evaluate taylor series??
  6. Oct 10, 2015 #5


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    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.

  7. Oct 10, 2015 #6
    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??
  8. Oct 10, 2015 #7
    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,

    [tex] \sum_{n=0}^{\infty} {\frac{(-1)^{\frac{n}{2}} z^n}{n!}} [/tex]
    Last edited: Oct 10, 2015
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