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Math Help - Possible to analytically find the sine of any angle?

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    Possible to analytically find the sine of any angle?

    Is it possible to find an exact answer for something like sine(46 degrees)? I understand how to use the sum formula for sine(a+b) to get something like sine(75). I also see how you could use the sine(a/2) formula to get something like sine (22.5) and keep halving it to get lower and lower.
    But if one only knows exact values for the sine of 30,45, and 60, the sum formula and the halving formula, wouldn't it be impossible to find an exact answer for the sine(23)?
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    Quote Originally Posted by lamp23 View Post
    Is it possible to find an exact answer for something like sine(46 degrees)? I understand how to use the sum formula for sine(a+b) to get something like sine(75). I also see how you could use the sine(a/2) formula to get something like sine (22.5) and keep halving it to get lower and lower.
    But if one only knows exact values for the sine of 30,45, and 60, the sum formula and the halving formula, wouldn't it be impossible to find an exact answer for the sine(23)?
    With only that knowledge, yes.
    You might want to check out the power series for sine (cosine).
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    Quote Originally Posted by lamp23 View Post
    Is it possible to find an exact answer for something like sine(46 degrees)? I understand how to use the sum formula for sine(a+b) to get something like sine(75). I also see how you could use the sine(a/2) formula to get something like sine (22.5) and keep halving it to get lower and lower.
    But if one only knows exact values for the sine of 30,45, and 60, the sum formula and the halving formula, wouldn't it be impossible to find an exact answer for the sine(23)?
    This is exactly the reason why Taylor Polynomials were created. If you think about a calculator, all that it can do is add (and by extension, subtract, multiply, divide and exponentiate). So when programming a calculator to evaluate (say) the sine of any angle, it needs to be programmed with some kind of polynomial (in other words, some combination of addition, subtraction, multiplication, division and exponentiation). I suggest you google "Taylor Series" for more information.
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    Is the graph of sin x based upon its series then or is there another theorem in calculus that would allow you to graph it?
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    Quote Originally Posted by Prove It View Post
    This is exactly the reason why Taylor Polynomials were created. If you think about a calculator, all that it can do is add (and by extension, subtract, multiply, divide and exponentiate). So when programming a calculator to evaluate (say) the sine of any angle, it needs to be programmed with some kind of polynomial (in other words, some combination of addition, subtraction, multiplication, division and exponentiation). I suggest you google "Taylor Series" for more information.
    Calculators do not in general use Taylor series representations of special functions, they use polynomial approximation over some range (which are not Taylor polynomials for the function but something like a least absolute error approximation on the range) and reduction formulae to reduce an argument to that range.

    You can also Google for CORDIC algorithms which is another approach.

    CB
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    Grand Panjandrum
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    Quote Originally Posted by lamp23 View Post
    Is the graph of sin x based upon its series then or is there another theorem in calculus that would allow you to graph it?
    Please post your real problem

    CB
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    Quote Originally Posted by CaptainBlack View Post
    Please post your real problem

    CB
    It's not homework. I just realized my textbook just fills in the entire graph without explaining if it's possible to find exact values or if the graph has to be approximated.
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    Grand Panjandrum
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    Quote Originally Posted by lamp23 View Post
    It's not homework. I just realized my textbook just fills in the entire graph without explaining if it's possible to find exact values or if the graph has to be approximated.
    But what does that mean? You don't need "exact" values to plot a graph, you plot a few points (and you only need a few significant figures for that, not an exact value) and connect them with a smooth curve.

    CB
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