Finding the explicit solution of a trig equation

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  • #1
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Hello,

I understand that solving an equation like f(x)=0 means finding those values of x that make the equality true.

In the case of some difficult trig equations (like sin(x)+tan(3x+2)+cos^2(x)=5 which I just made it) it may not be possible to find an explicit solution. What does that mean? Why is exactly a solution that is called explicit? Is there an implicit solution?

I would say that an explicit solution gives the exact x values that render the equation true...

thanks!
 

Answers and Replies

  • #2
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##sin(x)+tan(3x+2)+cos^2(x)-5=0## is the implicit solution, as it determines the set of values ##x## which makes the equation true. An explicit solution would be something like ##x= z\cdot \dfrac{\pi}{4}\; , \;z \in \mathbb{Z}##. This is easy for functions like sine and cosine, and presumably impossible for those of your equation. In such a case one has to compute numerically solutions, which are by nature not exact nor can you compute all of them.
 
  • #3
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The function (interpreted as f(x)=0) is periodic with a period of 2 pi, in this case you can numerically calculate all solutions in 0 to 2 pi and all others are an integer multiple of 2 pi larger/smaller. There are equations where this doesn’t work, however.
 
  • #4
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Thank you fresh_42. I guess so since all the trig identities I tried don't get me close to a form where I can take an inverse function and solve for x. That means explicit solutions does not exist. If x was small, I could approximate the various terms to polynomials. Butt no if x can have any value....

Why is the equation equal to zero the implicit solution? All I see is an equation set equal to zero with various possible x values as solutions...
 
  • #5
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Thanks mfb. Numerical is the way to go unless there are fancy substitution tricks or sophisticated approaches to manipulate the various trig terms into something manageable.
 
  • #6
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If x was small, I could approximate the various terms to polynomials.
That is still possible but it wouldn't help here - you would need to many terms and still get something that has to be solved numerically.
Why is the equation equal to zero the implicit solution? All I see is an equation set equal to zero with various possible x values as solutions...
That is exactly what an implicit solution means. You have an equation (more generally, a set of equations) that has to be satisfied for solutions.
 
  • #7
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Why is the equation equal to zero the implicit solution?
##A(x)=5## is as good as ##A(x)-5=0## is, there is no difference. It is just a convention to write them as ##\ldots =0## because these roots (values of ##x## with ##\ldots =0##) are one of the first things you want to know about a function.
 

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