Solve for x y=sin(x-pi)

Problem

y=sin(x−π)

Solution

1. Isolate the sine term by observing the equation is already in the form y=sin(u) where u=x−π

2. Apply the inverse sine function to both sides of the equation to begin solving for the variable inside the argument.

arcsin(y)=x−π

3. Isolate x by adding π to both sides of the equation.

x=arcsin(y)+π

4. Account for periodicity and multiple solutions if x is not restricted to a specific interval. The general solution for the sine function involves two branches and an integer k

x=π+(−1)^k*arcsin(y)+k*π

5. Simplify the expression using the identity sin(x−π)=−sin(x) This means y=−sin(x) or sin(x)=−y

x=arcsin(−y)

6. Apply the odd property of the arcsine function, arcsin(−y)=−arcsin(y) to find the principal value.

x=π−arcsin(y)

Final Answer

x=π+arcsin(y)

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