Find a root for this expression:

In summary, the equation cos(x)+cos(ix)+cos(x*i^3/2)+cos(x*i^1/2)=0 has been extensively explored, but no analytic root has been found. Some approximations have been calculated and a potential complex root has been identified, but there is no known simplification for this equation.
  • #1
MaxwellPhill
4
0
cos(x)+cos(ix)+cos(x*i^3/2)+cos(x*i^1/2)=0 for x

I have spent a lot of time finding an analytic root to this equation without success. An analytic root may not exist. I don't know. It is roughly equal to (8facorial)^(1/8)
 
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  • #2
I don't think there is an anaylitical solution and I am not sure that there even is a root.

x = 0 is closer than your x = 3.764350600, but that's only taken it down to 4 rather than 8.0003108.
 
  • #3
Hmm..I ust realized for the expression above the root must be complex of the form i^1/4.

Trig functions of the form cosx+cosy =0 can be solved because of the product formula:

cos(x)+cox(y)=2*(cos((x+y)/2))*(cos((x-y)/2)))

but is there such a formula for cos(x)+cos(y)+cos(z) ?
 
  • #4
I did some 3D Plots in mathematica to get a rough idea where a root might be and then used the FindRoot function and got that 1.4405686011239758 - 3.477840254362339i is approximatly a root.

I don't think there is any simplification like you want.
 

FAQ: Find a root for this expression:

What is a root in mathematics?

A root in mathematics refers to a number that, when multiplied by itself a certain number of times, equals the original number. For example, the square root of 25 is 5 because 5 multiplied by itself (5 x 5) equals 25.

Why do we need to find a root for an expression?

Finding a root for an expression allows us to solve for the unknown variable in an equation. It helps us understand the relationship between different quantities and can be used to solve real-world problems.

How do you find a root for an expression?

To find a root for an expression, you can use various mathematical methods such as factoring, the quadratic formula, or trial and error. The specific method used will depend on the complexity of the expression.

Can every expression have a root?

No, not every expression has a root. Some expressions may have complex roots (involving imaginary numbers) or no real roots at all. It is important to determine the type of roots an expression has before attempting to find them.

What are some real-world applications of finding roots for expressions?

Finding roots for expressions can be used in various fields such as engineering, economics, and physics. For example, in engineering, finding roots can help determine the optimal dimensions of a structure or the best design for a machine. In economics, finding roots can be used to analyze supply and demand equations. In physics, finding roots can help solve problems related to motion and forces.

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