The solution to cos x = 2 or any number > 1

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In summary: So the solution is x = - ln(2 + √3) i or in polar form, x = - ln(2 + √3) e^{iπ/2}.In summary, the inverse cosine of any number greater than 1 has a solution of x = - ln(2 + √3) i or in polar form x = - ln(2 + √3) e^{iπ/2}, obtained by using the identity (eiθ+e-iθ)/2 = cosθ and solving for eiθ.
  • #1
smutangama
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Homework Statement



Generally the inverse cosine of any number > 1




Homework Equations



cos x = 2

The Attempt at a Solution



Obviously by putting this in a calculator, you get an error so the root has to be complex

I used the identity (e+e-iθ)/2 = cosθ

through a bit of manipulation, I came to e = 2±√3

with the solutions being x = ln(2±√3) / i

is this correct?

thanks
 
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  • #2
Watch out with your calculations, to keep them clear for yourself and others.
But, your values for x are indeed correct.
 
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  • #3
smutangama said:

Homework Statement



Generally the inverse cosine of any number > 1




Homework Equations



cos x = 2

The Attempt at a Solution



Obviously by putting this in a calculator, you get an error so the root has to be complex

I used the identity (e+e-iθ)/2 = cosθ

through a bit of manipulation, I came to e = 2±√3

with the solutions being x = ln(2±√3) / i

is this correct?

thanks
Correct so far but you have not finished. You should be able to get the answer in eityer of the
"standard forms" a+ bi or [tex]re^{i\theta}[/tex]. First, dividing by i is the same as multiplying by -i so this is [itex]x= -ln(2\pm \sqrt{3})i[/itex]. Next, ln(x) where x is a NEGATIVE real number is itself complex.
 
  • #4
HallsofIvy said:
ln(x) where x is a NEGATIVE real number is itself complex.
True, but 2 - √3 > 0.
 

FAQ: The solution to cos x = 2 or any number > 1

1. What does "cos x = 2" mean?

When we say "cos x = 2", it means that the cosine of the angle x is equal to 2. In other words, if we were to plug in the value of x into the cosine function, we would get 2 as the result.

2. Is there a specific value for x that satisfies "cos x = 2"?

Yes, there is a specific value for x that satisfies this equation. In this case, x is approximately 0.583 radians, or 33.17 degrees. However, it is important to note that there are infinitely many values of x that satisfy this equation, as cosine is a periodic function.

3. Can we solve "cos x = 2" algebraically?

No, we cannot solve "cos x = 2" algebraically. This is because cosine is a transcendental function, meaning it cannot be expressed as a finite combination of algebraic operations. Therefore, we must use numerical methods to approximate the solution.

4. How can we use a graph to find the solution to "cos x = 2"?

We can plot the graph of the cosine function and the line y = 2. The points where these two graphs intersect will give us the solutions to the equation "cos x = 2". In this case, there will be two points of intersection, as cosine is a periodic function that repeats itself.

5. What is the significance of "cos x = 2" in real-world applications?

The cosine function is commonly used in fields such as physics, engineering, and mathematics. It represents the relationship between the sides of a right triangle and the angles. In real-world applications, "cos x = 2" may represent the magnitude of a force acting on an object at a certain angle, or the amplitude of a wave with a specific frequency.

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