Solve for x sin(x/2-pi/4)=1/2

Problem

sin(x/2−π/4)=1/2

Solution

1. Identify the general solution for the sine function. For an equation sin(θ)=a the solutions are θ=arcsin(a)+2*k*π and θ=π−arcsin(a)+2*k*π where k is any integer.

2. Determine the reference angles for sin(θ)=1/2 The primary values are π/6 and (5*π)/6

3. Set up the two equations for the argument of the sine function.

x/2−π/4=π/6+2*k*π

x/2−π/4=(5*π)/6+2*k*π

4. Isolate the term containing x in the first equation by adding π/4 to both sides.

x/2=π/6+π/4+2*k*π

x/2=(5*π)/12+2*k*π

5. Solve for x in the first case by multiplying the entire equation by 2

x=(5*π)/6+4*k*π

6. Isolate the term containing x in the second equation by adding π/4 to both sides.

x/2=(5*π)/6+π/4+2*k*π

x/2=(13*π)/12+2*k*π

7. Solve for x in the second case by multiplying the entire equation by 2

x=(13*π)/6+4*k*π

Final Answer

sin(x/2−π/4)=1/2⇒x=(5*π)/6+4*k*π,x=(13*π)/6+4*k*π

Want more problems? Check here!