Zeros Problem of a trigonometric function

[ISTER_REG]

Hi,

Given is the following equation [MATH]cos(x) +sin(x) = 0[/MATH], my first conversion is [MATH]cos(x) = -sin(x)[/MATH]. I know that it is possible to re-write the -sin(x) term. Therefore [MATH]-sin(x) = cos(x + \frac{\pi}{2})[/MATH]. Now I would like to apply [MATH]cos^{-1}(x)[/MATH] to the equation [MATH]cos(x) = cos(x + \frac{\pi}{2})[/MATH]. This brings me to [MATH]x = x + \frac{\pi}{2}[/MATH] which is clearly not true since [MATH]0 \neq \frac{\pi}{2}[/MATH].

But if I use [MATH]cos(x) = cos(-x)[/MATH] in the above equation [MATH]cos(x) = cos(x + \frac{\pi}{2})[/MATH] so [MATH]cos(-x) = cos(x + \frac{\pi}{2})[/MATH] it comes to [MATH]-2x =\frac{\pi}{2}[/MATH], knowing that cosine is a [MATH]2\pi[/MATH] periodic function I would say the solution (based on the last abstract) is [MATH]x = -\frac{\pi}{4} \pm n\pi[/MATH]. Since [MATH]-2x = \frac{\pi}{2} \pm 2n\pi[/MATH]
Why does the first part of the invoice not work? Why can't I just use [MATH]cos^{-1}(x)[/MATH] for [MATH]cos(x) = cos(x + \frac{\pi}{2})[/MATH], but I have to consider that [MATH]cos(x) = cos(-x)[/MATH] is valid (which leads to the right solution). So why do I get a correct solution in the second paragraph, while I get no solution in the first paragraph?

Thanks for coming answers!

 

[Steven G]

If cos(A) = cos(B) it does NOT follow that A=B. Why do you think that? Cos(x) is periodic!

For example, cos(0) = cos(2pi) = cos(4pi)= cos(6pi) =...=cos(-2pi)=cos(-4pi) = cos(-6pi) =... even though the angles are all different!

Please try again.

 

[Steven G]

Look here

 

[ISTER_REG]

Hi thanks for your answer,

Why I believe this, I once learned that the cosine is a [MATH]2\pi[/MATH] periodic function, but the following applies: [MATH]cos(x) = cos(x + 2n\pi)[/MATH]
On the other hand I must say, does your example with cos(A) = cos(B) already make sense somewhere.

 

[Steven G]

[MATH]Yes,\: \cos(x) = \cos(x + 2n\pi)\: but\: x \neq x+2n\pi\:unless\: n=0[/MATH]

 

[ISTER_REG]

Thanks, that made it clear for me!