Finding the explicit solution of a trig equation

[fog37]

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]

$sin(x)+tan(3x+2)+cos^2(x)-5=0$ 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]

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.

 

[fog37]

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...

 

[fog37]

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]

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]

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.