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

Problem

sin(x)=−√(,2)/2

Solution

1. Identify the reference angle by finding the value of θ in the first quadrant where sin(θ)=√(,2)/2

θ=π/4

2. Determine the quadrants where the sine function is negative. Since sin(x) is negative, x must be in Quadrant III or Quadrant IV.

3. Calculate the angle in Quadrant III by adding the reference angle to π

x=π+π/4

x=(5*π)/4

4. Calculate the angle in Quadrant IV by subtracting the reference angle from 2*π

x=2*π−π/4

x=(7*π)/4

5. Generalize the solution to include all possible rotations by adding 2*π*n where n is any integer.

x=(5*π)/4+2*π*n

x=(7*π)/4+2*π*n

Final Answer

sin(x)=−√(,2)/2⇒x=(5*π)/4+2*π*n,(7*π)/4+2*π*n

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