Sine division formula

[macrofire]

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.

[peachoasis]

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.

[aldld]

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.

[Zenith]

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.

[macrofire]

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...

[Rahul]

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.