# Finding the explicit solution of a trig equation

• I
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!

fresh_42
Mentor
##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.

mfb
Mentor
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.

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...

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.

mfb
Mentor
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.

fresh_42
Mentor
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.