macrofire Posted October 11, 2012 Report Share Posted October 11, 2012 Say you have Sin X and Sin Y, where X and Y are plane angles in degrees. Is there a way to simplify (Sin X)/(Sin Y)?I know this works: cis(x)/cis(y) but I'm hoping to find an analog in the conventional trig functions. 1 Reply Link to post Share on other sites More sharing options...
peachoasis Posted December 23, 2012 Report Share Posted December 23, 2012 There is no simple formula for sin(x)/sin(y).The reason cis(x)/cis(y)=cis(x-y) is due to De Moivre's theorem.We can rewrite cis(x)/cis(y) as cis(x)*(cis(y))^(-1).By De Moivre's theorem, this is the same as cis(x)*cis(-y).Using some trig properties and identities, you arrive at the conclusion: cis(x)*cis(-y)=(cosx+isinx)(cosy-isiny)=(cosxcosy+sinxsiny)+i(sinxcosy-sinycosx)=cos(x-y)+isin(x-y)=cis(x-y)De Moivre's theorem applies only to complex numbers so if just have sinx and siny, you cannot simplify it in an analogous manner. Reply Link to post Share on other sites More sharing options...
aldld Posted December 23, 2012 Report Share Posted December 23, 2012 sin(x)/sin(y) = sin(x)csc(y). Although that's not helpful at all I doubt there's any elegant formula, though sin(x)/sin(y) is already pretty simple as it is. Reply Link to post Share on other sites More sharing options...
Zenith Posted January 5, 2013 Report Share Posted January 5, 2013 sin(x)/sin(y) can be written/simplified in a number of ways:Firstly, you may reorganize the phrase as aldld did above into sin(x)csc(y), however this is merely a slightly different name for the same thing.Another form is "(e^(-i x)-e^(i x))/(e^(-i y)-e^(i y))", although this is also simply a substitution rearrangement of the same mathematical phrase.You may also assume that the x and y values are real (as in real numbers), and given that, change the form to: "-(2sin(x) sin(y)) / (cos(2y)-1)"Hope this helps P.S. The 3D graph for this formula is attached as a JPEG file so take note of that if you think visuals would help you. Reply Link to post Share on other sites More sharing options...
macrofire Posted January 9, 2013 Author Report Share Posted January 9, 2013 sin(x)/sin(y) can be written/simplified in a number of ways:Firstly, you may reorganize the phrase as aldld did above into sin(x)csc(y), however this is merely a slightly different name for the same thing.Another form is "(e^(-i x)-e^(i x))/(e^(-i y)-e^(i y))", although this is also simply a substitution rearrangement of the same mathematical phrase.You may also assume that the x and y values are real (as in real numbers), and given that, change the form to: "-(2sin(x) sin(y)) / (cos(2y)-1)"Hope this helps P.S. The 3D graph for this formula is attached as a JPEG file so take note of that if you think visuals would help you.Thanks, although...you made it somewhat more complicated. Really, you gave me 3 identities to work with...I just wanted some really simple identity though... Reply Link to post Share on other sites More sharing options...
Rahul Posted January 9, 2013 Report Share Posted January 9, 2013 sin(x)/sin(y) can be written/simplified in a number of ways:Firstly, you may reorganize the phrase as aldld did above into sin(x)csc(y), however this is merely a slightly different name for the same thing.Another form is "(e^(-i x)-e^(i x))/(e^(-i y)-e^(i y))", although this is also simply a substitution rearrangement of the same mathematical phrase.You may also assume that the x and y values are real (as in real numbers), and given that, change the form to: "-(2sin(x) sin(y)) / (cos(2y)-1)"Hope this helps P.S. The 3D graph for this formula is attached as a JPEG file so take note of that if you think visuals would help you.Thanks, although...you made it somewhat more complicated. Really, you gave me 3 identities to work with...I just wanted some really simple identity though...There's no way to really simplify this as with cis(x)/cis(y)=cis(x-y); all that's being given is identities and different ways of expressing this phrase. A simple identity does not exist as far as I know to simplify this further. Maybe with taylor series? I'll look into it. Reply Link to post Share on other sites More sharing options...
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