Proof that if cos(x+y)=cos(x), then y=2*pi*k?

[suckmyfish1]

Recently I remembered a property of cosine that I learned in high school, namely that the cosine function repeats every 2$\pi$k times for every integer k. I was told that naturally, then, if cos(x+y)=cos(x), then y is of the form 2$\pi$k, where k is an integer. However, I was looking for a formal proof of this. Can someone help me out? Thanks.

 

[micromass]

What is your definition of cosine and of pi?? What can we assume known??

 

[Ferramentarius]

Recently I remembered a property of cosine that I learned in high school, namely that the cosine function repeats every 2$\pi$k times for every integer k. I was told that naturally, then, if cos(x+y)=cos(x), then y is of the form 2$\pi$k, where k is an integer. However, I was looking for a formal proof of this. Can someone help me out? Thanks.

The described implication does not hold exclusively because $cos(x) = cos(-x) = cos(x-2x)$ for any x. As $cos(z) = cos(w)$ only when $z$ and $w$ are at the same x-coordinate of the unit circle we have $z = \pm w + 2\pi k$. In the quoted case it is $x = \pm (x+y) \rightarrow y = 2\pi k \vee y = -2x + 2\pi k$.

 

[Someone2841]

Recently I remembered a property of cosine that I learned in high school, namely that the cosine function repeats every 2$\pi$k times for every integer k. I was told that naturally, then, if cos(x+y)=cos(x), then y is of the form 2$\pi$k, where k is an integer. However, I was looking for a formal proof of this. Can someone help me out? Thanks.

$cos(a) = cos(b) \text{ iff } a = b + 2\pi k \text{ or } a = -b + 2\pi k$

Therefore, $cos(x) = cos(x+y) \text{ iff } x = (x + y) + 2\pi k \text{ or } x = -(x+y) + 2\pi k$.

This reduces to $y = + 2\pi k \text{ or } y = -2x + 2\pi k$.

This proves that your friend's statement was incomplete, as y could be in the form of -2x+2πk as well

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Edit: Although, if one considers y as a constant (as opposed to a function of x), then y could only be in the form of 2πk and work for all values of x. Maybe your friend was on the something after all :p

 

[blue_raver22]

The statement that cos(x+y)=cos(x) implies that y=2*pi*k is a well-known property of the cosine function and can be proven using the fundamental trigonometric identities.

First, we can rewrite cos(x+y) as cos(x)cos(y)-sin(x)sin(y) using the sum formula for cosine. Then, equating this to cos(x), we get:

cos(x)cos(y)-sin(x)sin(y)=cos(x)

Rearranging this equation, we get:

cos(x)cos(y)=cos(x)+sin(x)sin(y)

Now, we can use the identity cos^2(x)+sin^2(x)=1 to substitute in for cos^2(x) and sin^2(x) in the above equation, giving us:

cos(x)(cos(y)-1)=sin(x)sin(y)

Next, we can divide both sides by cos(x) to get:

cos(y)-1=tan(x)sin(y)

Finally, we can use the fact that cos(y) repeats every 2*pi times to show that if cos(y)-1=tan(x)sin(y), then y must be of the form 2*pi*k, where k is an integer. This can be done by considering the different values of y that satisfy the equation for different values of k.

Therefore, we can conclude that if cos(x+y)=cos(x), then y=2*pi*k, where k is an integer. This is a formal proof of the property that was mentioned in the content.