CTK Exchange CTK Wiki Math Front Page Movie shortcuts Personal info Awards Terms of use Privacy Policy Cut The Knot! MSET99 Talk Games & Puzzles Arithmetic/Algebra Geometry Probability Eye Opener Analog Gadgets Inventor's Paradox Did you know?... Proofs Math as Language Things Impossible My Logo Math Poll Other Math sit's Guest book News sit's Recommend this site         CTK Exchange

 Subject: "how do I solve this trigonometric equations system?" Previous Topic | Next Topic
 Conferences The CTK Exchange College math Topic #710 Printer-friendly copy Email this topic to a friend Reading Topic #710
Manuel Sandoval guest
Mar-28-09, 03:13 PM (EST)

"how do I solve this trigonometric equations system?"

 Hello:I am trying to solve this system:A*cos(x) + B*sin(y) = CD*sin(x) + E*cos(y) = FI have tried several substitutions, but it will always lead to a 4th degree equation. I habe also used identities, for example sin(y) = sqrt(1-cos(y)^2), sin(x)=(exp(i*x)-exp(i*x))/2*i and cos(x)=(exp(i*x)+exp(i*x))/2, and so on...For instance: A*cos(x) + B*sin(y) = C => sin(y)^2 = ((C-A*cos(x))/B)^2D*sin(x) + E*cos(y) = F => cos(y)^2 = ((F-D*sin(x))/E)^2 = 1- sin(y)^2So ((F-D*sin(x))/E)=^2 = 1 - ((C-A*cos(x))/B)^2When replacing sin(x) = sqrt(1-cos(x)^2 and expanding, I will get a 4th degree equation.HOWEVER: I remember that in highschool I used to solve this without having to solve a 4th degree equation. I do know the solution for a 4th degree equation, but as you may know, it's a BIG formula.The solution for the system is:y1= arcsin((M*G+K)/(G^2+H^2))y2= arcsin((M*G-K)/(G^2+H^2))x1= arccos((F-E*y1)/d)x2= arccos((F-E*y2)/d)Where M=C^2+B^2+F^2-A^2, G=2*F*E, H=-2*B*CI got it like 10 years ago! but now I can't remember the trick. I need to document a program I wrote for solving equations.Thanks!

Gerenuk guest
Jul-10-10, 06:58 AM (EST)

1. "RE: how do I solve this trigonometric equations system?"
In response to message #0

 Oh, I was just trying hard to solve the same problem.The linkhttps://www.geometrictools.com/Documentation/IntersectionOfEllipses.pdfseemed to indicate that you cannot avoid a 4th order polynomial.So what you've written down is the actual solution to the problem???That would be amazing.I've tried calculations with the discriminant of the 4th order polynomial but it doesn't quite work yet.

Gerenuk guest
Jul-10-10, 06:58 AM (EST)

2. "RE: how do I solve this trigonometric equations system?"
In response to message #0

 What's the variables "d" and "K" anyway?

 Conferences | Forums | Topics | Previous Topic | Next Topic
 Select another forum or conference Lobby The CTK Exchange (Conference)   |--Early math (Public)   |--Middle school (Public)   |--High school (Public)   |--College math (Public)   |--This and that (Public)   |--Guest book (Protected)   |--Thoughts and Suggestions (Public) Educational Press (Conference)   |--No Child Left Behind (Public)   |--Math Wars (Public)   |--Mathematics and general education (Public) You may be curious to have a look at the old CTK Exchange archive.  