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anonymous
 one year ago
what are all the solutions to cos(x) = cos(y) ?
anonymous
 one year ago
what are all the solutions to cos(x) = cos(y) ?

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0implifying cos(x) = 1cos(y) Multiply cos * x cosx = 1cos(y) Multiply cos * y cosx = 1cosy Solving cosx = 1cosy Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add 'cosy' to each side of the equation. cosx + cosy = 1cosy + cosy Combine like terms: 1cosy + cosy = 0 cosx + cosy = 0 Factor out the Greatest Common Factor (GCF), 'cos'. cos(x + y) = 0 Subproblem 1 Set the factor 'cos' equal to zero and attempt to solve: Simplifying cos = 0 Solving cos = 0 Move all terms containing c to the left, all other terms to the right. Simplifying cos = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. Subproblem 2 Set the factor '(x + y)' equal to zero and attempt to solve: Simplifying x + y = 0 Solving x + y = 0 Move all terms containing c to the left, all other terms to the right. Add '1x' to each side of the equation. x + 1x + y = 0 + 1x Combine like terms: x + 1x = 0 0 + y = 0 + 1x y = 0 + 1x Remove the zero: y = 1x Add '1y' to each side of the equation. y + 1y = 1x + 1y Combine like terms: y + 1y = 0 0 = 1x + 1y Simplifying 0 = 1x + 1y The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@MasterGeen no copy and paste please

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0YOU WANTED AN ANSWER RIGHT LOL

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0cos(x)=cos(y)>x=2kpi+y and x=2kpiy

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@amirreza1870 how did you get that?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Cos and Sin have period of 2pi, i.e. cos(x+2pi)=cos(x) for exemple. WIth that in mind, cos(x) = cos(y) then \(x = y+2n\pi, n\in N\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@M4thM1nd what about the other solution?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the period of cos(x) is 2pi for example cos60=cos420 so the first answer was 2kpi+y then you know that cos(x)=cos(x) so the second answer is 2kpiy.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0another solution is due the fact that Cos is an even function, i.e. cos(x) = cos(x). So... \(x = y+2n\pi, n\in N\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok, make sense. But isn't it more like "insight" than actual computation?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0m4th mind helping me after?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0For a more stepbystep solution, cos(x) = cos(y) x = acos(cos(y)) \(x=\pm y+2n\pi\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0but inverse cosine can only gives a value between [0,pi]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I missed the solution x = y + (2pi)n initially because didn't realize cos(x) = cos(x).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes, the range of acos(x) is [0, pi], but you get to keep in mind that cos is positive in the first and fourth quadrant and negative in second and third. So, if you evaluate acos(cos(3pi/2)) you will get 3pi/4 for example. This means we have a "refletion" around x = pi, for possible solutions

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok. Makes sense. Thank you
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