Finding trigonometric solution to a cubic equation using computer

[Seydlitz]

Hello,

Homework Statement

I get this question from Mathematical Methods by Boas page 74 problem 25. The question states:

"Use a computer to find the three solutions of the equation ##x^3-3x-1=0##. Find a way to show that the solutions can be written as ##2cos(\frac{\pi}{9})##, ##-2cos(\frac{2\pi}{9})##, ##2cos(\frac{4\pi}{9})##.

The Attempt at a Solution

Again I'm still confused on what she means by use a computer?

Can I just use Wolfram Alpha to find the solution in exact form and then show it that it can be expressed to polar form and hence trigonometric function? I checked with WA and it gives me the solution x in rectangular form. I just convert that to trigonometric form and it does get the same answer as the question.

But then on the other hand, the question itself was probably written long before program such as WA existed. Should I just skip this kind of problem next time?

Thank You

 

[mighty2000]

Thank you for your question. It is understandable to be confused about using a computer to solve this problem. In this case, the use of a computer is simply to aid in finding the numerical solutions to the equation. This can be done by using a numerical solver, such as Wolfram Alpha, as you have suggested. However, it is important to also understand the underlying mathematical concepts and methods used to solve the equation.

In this case, the solutions to the equation can be expressed in terms of trigonometric functions, as you have mentioned. This is because the equation can be rewritten as ##x^3-3x-1=(x-2cos(\frac{\pi}{9}))(x+2cos(\frac{2\pi}{9}))(x-2cos(\frac{4\pi}{9}))##. This is known as the trigonometric factorization method, which can be used to solve cubic equations involving trigonometric functions.

So, while it is useful to use a computer to find the numerical solutions, it is also important to understand the underlying mathematical concepts and methods. I would suggest practicing solving similar problems by hand to strengthen your understanding. And no, you should not skip this type of problem, as it is important to have a solid understanding of different problem-solving methods in mathematics.

I hope this helps clarify things for you. Good luck with your studies!