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

Problem

sin(2*x)=−√(,2)/2

Solution

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

2. Determine the quadrants where the sine function is negative. Sine is negative in the third and fourth quadrants.

3. Find the general solutions for the argument 2*x by applying the quadrant rules.

2*x=π+π/4+2*π*k

2*x=2*π−π/4+2*π*k

4. Simplify the expressions for 2*x

2*x=(5*π)/4+2*π*k

2*x=(7*π)/4+2*π*k

5. Solve for x by dividing all terms by 2

x=(5*π)/8+π*k

x=(7*π)/8+π*k

Final Answer

sin(2*x)=−√(,2)/2⇒x=(5*π)/8+π*k,(7*π)/8+π*k

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