Need help in solving Trigonometric Transcendental equations.

[cannibol_90]

Hi,

I would like to know how to solve for these set of equations.

A = B sin(2.(Q1)) + C tan(Q2) and
D = (1/C) sin(2.(Q2)) + (1/B) tan(Q1)

where A,B,C and D are integer values while Q1 and Q2 are the unknowns.
Please Help!

 

[albbg]

I guess you know that:

[tan(a)]^2 = 1 / [sin(2a)]^2 - 1

Are you sure ? let a=0

[tan(0)]^2=0
sin(0)=0 then 1 / [sin(0)]^2 - 1->infinite

developing instead [tan(a)]^2:

[tan(a)]^2=[sin(a)]^2/[cos(a)]^2, but

[sin(a)]^2=[1-cos(2a)]/2
[cos(a)]^2=[1+cos(2a)]/2

then:

[tan(a)]^2=[1-cos(2a)]/[1+cos(2a)]

 

[KerimF]

Thank you albbg.

I deduced my silly formula when in bed yesterday to help me sleep. And to verify it, I replaced 'a' with 30 deg. So when I found out it is ok (bad luck), I had a good sleep
Next day, I didn't bother myself to check it more carefully since I used having confidence in my brain. But I should remember from now on that I am not young anymore at age 64
Thank you again for pointing out my silly mistake here.
I wished that Cannibol found it first.
Knowing for sure that something is wrong is a good sign for one's ability in finding the right answer.

 

[cannibol_90]

Ok, Let me be more simple!

5 = 3sin(2A) + 4tan(B)
7 = (1/4)sin(2B) + (2/3)tan(A)

Now, can anyone help me solve these equations, please?

 

[albbg]

It's quite complex to solve the system of 2 equations you propose, unless you use a numerical solver.

however, since sin(2*A)=2*sin(A)*cos(A) then

5=6*sin(A)*cos(A)+4*sin(B)/cos(B)
7=1/2*sin(B)*cos(B)+2/3*sin(A)/cos(A)

now we know that cos(A)=sqrt[1-sin(A)^2] thus if we call x=sin(A) then cos(A)=sqrt(1-x^2)
and, of course if y=sin(B) then cos(B)=sqrt(1-y^2)

5=6*x*sqrt(1-x^2)+4*y/sqrt(1-y^2)
7=1/2*y*sqrt(1-y^2)+2/3*x/sqrt(1-x^2)

You can try to solve these last two

 

[Ratch]

Hi,

I would like to know how to solve for these set of equations.

A = B sin(2.(Q1)) + C tan(Q2) and
D = (1/C) sin(2.(Q2)) + (1/B) tan(Q1)

where A,B,C and D are integer values while Q1 and Q2 are the unknowns.
Please Help!

As shown below, graph the two equations and find the solution where the graphs intersect.



Ratch

 

[cannibol_90]

Ok Guys, I came up with a way to solve the above equations.

It's a bit lengthy one, but trust me this works!

A = B sin(2.(Q1)) + C tan(Q2) and
D = (1/C) sin(2.(Q2)) + (1/B) tan(Q1)

Rearrange the first equation to obtain Q1 and the second one to obtain Q2.

Q1 = (sin^-1 ((A-Ctan(Q2))/B))*(1/2)
Q2 = (sin^-1 (C*(D-((1/B)tan(Q1))))

Now, assume you have the initial values of Q1 and Q2.

Now substitute the initial value of Q2 in the first equation and obtain Q1.
The obtained value of Q1 is put into the 2nd equation to obtain Q2.
This completes the 1st iteration.

Now again re-substitute the obtained values (new) of Q1 and Q2 into the above equations to obtain 2 new solutions for Q1 and Q2. This completes the 2nd iteration.

Keep on iterating until the values of Q1 and Q2 become stable, by which I mean they don't change anymore. This is the final solution for Q1 and Q2.

I have tried it out and the answer is coming to be the exact one as mentioned in the text book.

Phew!

CASE CLOSED, PROBLEM SOLVED!

But still I am keeping this thread open, as one day maybe someone comes up with a unique way to solve it. Who knows?

I thank everyone for helping me out!

Have a nice day! Byeee...

 

[albbg]

Yes, it's a simple numerical method. Unless you proof it, you can't be sure it converges for any value of the parameters.

 

[FvM]

Do you say, your textbook already stated that the equations can't be analytically solved?

I had guessed this, too. But once you decide for an iterative (= numerical) solution, it's just number crunching. Finding good start values is helpful but not necessarily required.

Unless you proof it, you can't be sure it converges for any value of the parameters.

Most likely it doesn't converge for arbitrary parameter combinations.

 

[cannibol_90]

Do you say, your textbook already stated that the equations can't be analytically solved?

I had guessed this, too. But once you decide for an iterative (= numerical) solution, it's just number crunching. Finding good start values is helpful but not necessarily required.


Most likely it doesn't converge for arbitrary parameter combinations.

Yes, I think you are right! It may not converge for arbitrary values, which I remember from my old experiences learning "numerical analysis" as a subject in my graduate days.

Actually, there are a pair of equations that gives you the initial values of Q1 and Q2, which I forgot to tell you. Sorry, I am an amateur naa... forgive me, MY MISTAKE!!!

Saying that, I thought it wouldn't help to solve the above equations.

Anyways, thank you all for bearing with a stupid like me.

 

[cannibol_90]

One more question for all of you.

If I have an equation such as:-

A = sin(Bx)tan(Cx)-Gsin(Bx)+Ftan(Cx).

How can I isolate the variable 'x' (like x = blah blah blah, R.H.S being devoid of the variable 'x'); where A, B, C, F and G are constants.

- - - Updated - - -

Do you say, your textbook already stated that the equations can't be analytically solved?

Well, it doesn't say anything! That is why I approached you guys.