Solve for x y=cos(x-pi/2)

Problem

y=cos(x−π/2)

Solution

1. Isolate the cosine term by ensuring the equation is in the form cos(u)=y

y=cos(x−π/2)

2. Apply the inverse cosine function to both sides to begin solving for the argument of the cosine.

arccos(y)=x−π/2

3. Account for the periodicity of the cosine function, which repeats every 2*π*n for any integer n

x−π/2=±arccos(y)+2*π*n

4. Isolate x by adding π/2 to both sides of the equation.

x=π/2±arccos(y)+2*π*n

5. Simplify using identities if desired. Since cos(x−π/2)=sin(x) the equation is equivalent to y=sin(x) which leads to a more compact general solution.

x=arcsin(y)

Final Answer

x=arcsin(y)+2*π*n,x=π−arcsin(y)+2*π*n

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