mighty mean and dreadful identity problem

[allegansveritatem]

Here is the problem:
I tried this from the left side and got into a tangle. tried it from the right and ran into an even deader end. Here is a sample of my labors:

I recognized the numerator for the square to linear formulation in both numerator and denominator and tried to do something with that....but came to grief. What to do here?

 

[Dr.Peterson]

That's a creative idea, and you may have just lost some pieces (e.g. keeping 2v in places where it should be just v), but I didn't try following the details too closely.

I'd just use the double-angle formulas directly, e.g. sin(2v) = 2 sin(v)cos(v). Give that a try.

 

[Deleted member 4993]

Here is the problem:View attachment 18177
I tried this from the left side and got into a tangle. tried it from the right and ran into an even deader end. Here is a sample of my labors:
View attachment 18178
I recognized the numerator for the square to linear formulation in both numerator and denominator and tried to do something with that....but came to grief. What to do here?

\(\displaystyle \frac{1+cos(2\nu) +sin(2\nu)}{1-cos(2\nu)+sin(2\nu)}\)

\(\displaystyle = \frac{2cos^2(\nu) +2sin(\nu)*cos(\nu)}{2sin^2(\nu)+2sin(\nu)*cos(\nu)}\)

Now factorize numerator and denominator and you are there......

 

[pka]

Here is the problem:View attachment 18177
I tried this from the left side and got into a tangle. tried it from the right and ran into an even deader end. Here is a sample of my labors:
View attachment 18178
I recognized the numerator for the square to linear formulation in both numerator and denominator and tried to do something with that....but came to grief. What to do here?

\(\dfrac{1+\sin(2v)+\cos(2v)}{1+\sin(2v)-\cos(2v)}=\dfrac{1+2\sin(v)\cos(v)+[\cos^2(v)-1]}{1+2\sin(v)\cos(v)-[1-2\sin^2(v)]}\)
__________________________\(=\dfrac{\cos(v)[2\sin(v)+\cos(v)]}{\sin(v)[2\sin(v)+\cos(v)]}\)

 

[allegansveritatem]

That's a creative idea, and you may have just lost some pieces (e.g. keeping 2v in places where it should be just v), but I didn't try following the details too closely.

I'd just use the double-angle formulas directly, e.g. sin(2v) = 2 sin(v)cos(v). Give that a try.

Thanks. I will give that a try tomorrow.

 

[allegansveritatem]

\(\displaystyle \frac{1+cos(2\nu) +sin(2\nu)}{1-cos(2\nu)+sin(2\nu)}\)

\(\displaystyle = \frac{2cos^2(\nu) +2sin(\nu)*cos(\nu)}{2sin^2(\nu)+2sin(\nu)*cos(\nu)}\)

Now factorize numerator and denominator and you are there......

that looks like something that would work. I will check it out tomorrow. Thanks

 

[allegansveritatem]

\(\dfrac{1+\sin(2v)+\cos(2v)}{1+\sin(2v)-\cos(2v)}=\dfrac{1+2\sin(v)\cos(v)+[\cos^2(v)-1]}{1+2\sin(v)\cos(v)-[1-2\sin^2(v)]}\)
__________________________\(=\dfrac{\cos(v)[2\sin(v)+\cos(v)]}{\sin(v)[2\sin(v)+\cos(v)]}\)

fill in the bllank? haha. I see where you are headed. I will work it out tomorrow. Thanks.

 

[allegansveritatem]

so, I went at it again today and after a few false starts I managed to bring forth a QED thus:

I thank all contributors to the thread.I need all the help I can get.

 

[Steven G]

Good job! You really should have all angles listed as sin and cos have no meanings at all!

 

[allegansveritatem]

Good job! You really should have all angles listed as sin and cos have no meanings at all!

Right....I guess I was economising on ink.

 

[Deleted member 4993]

Right....I guess I was economizing on ink.

Remember the instructor can then economize on grades - and fight grade inflation.

 

[topsquark]

I own stock in red pen ink.

-Dan

 

[Dr.Peterson]

In scratch work, once there is only one argument, I commonly rewrite an equation like [MATH]\frac{2c^2 +2sc}{2s^2+2sc}[/MATH] to save pencil lead (I don't use ink ...). But before showing my work to anyone, I would translate it back to the usual form. (In principle, I could legally write what I do if I first defined [MATH]s = \sin(v)[/MATH] and [MATH]c = \cos(v)[/MATH], but even then I might get some criticism, so I avoid it.)

 

[allegansveritatem]

In scratch work, once there is only one argument, I commonly rewrite an equation like [MATH]\frac{2c^2 +2sc}{2s^2+2sc}[/MATH] to save pencil lead (I don't use ink ...). But before showing my work to anyone, I would translate it back to the usual form. (In principle, I could legally write what I do if I first defined [MATH]s = \sin(v)[/MATH] and [MATH]c = \cos(v)[/MATH], but even then I might get some criticism, so I avoid it.)

from this time forth I am going to try to clean up my posts...I might even learn to use math software.

 

[allegansveritatem]

Remember the instructor can then economize on grades - and fight grade inflation.

oh well, I'm not in this for grades.