Solve for x 2sin(x)+ square root of 2=0

Problem

2*sin(x)+√(,2)=0

Solution

1. Isolate the sine term by subtracting √(,2) from both sides of the equation.

2*sin(x)=−√(,2)

2. Divide both sides by 2 to solve for sin(x)

sin(x)=−√(,2)/2

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

(x_ref)=π/4

4. Determine the quadrants where the sine function is negative, which are Quadrant III and Quadrant IV.

x=π+π/4

x=2*π−π/4

5. Calculate the specific angles within the interval [0,2*π)

x=(5*π)/4

x=(7*π)/4

6. Generalize the solution by adding multiples of the period 2*π where n is any integer.

x=(5*π)/4+2*n*π

x=(7*π)/4+2*n*π

Final Answer

x=(5*π)/4+2*n*π,(7*π)/4+2*n*π

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