Solve for x sin(x/2)=( square root of 2)/2

Problem

sin(x/2)=√(,2)/2

Solution

1. Identify the reference angle for the sine function. The value √(,2)/2 corresponds to the sine of π/4 (or 45.

2. Determine the general solutions for the sine equation. Since sine is positive in the first and second quadrants, the argument x/2 must satisfy:

x/2=π/4+2*k*π

x/2=π−π/4+2*k*π=(3*π)/4+2*k*π

where k is any integer.

3. Solve for x by multiplying both sides of each equation by 2

x=2*(π/4+2*k*π)=π/2+4*k*π

x=2*((3*π)/4+2*k*π)=(3*π)/2+4*k*π

Final Answer

sin(x/2)=√(,2)/2⇒x=π/2+4*k*π,(3*π)/2+4*k*π

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