Solve for x sin(2x)=-( square root of 3)/2

Problem

sin(2*x)=−√(,3)/2

Solution

1. Identify the reference angle by finding the value of θ such that sin(θ)=√(,3)/2 The reference angle is π/3

2. Determine the quadrants where the sine function is negative. Sine is negative in Quadrant III and Quadrant IV.

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

Quadrant III: *2*x=π+π/3+2*k*π=(4*π)/3+2*k*π

Quadrant IV: *2*x=2*π−π/3+2*k*π=(5*π)/3+2*k*π

4. Solve for x by dividing the equations by 2

x=(2*π)/3+k*π

x=(5*π)/6+k*π

5. State the solution where k is any integer.

Final Answer

sin(2*x)=−√(,3)/2⇒x=(2*π)/3+k*π,(5*π)/6+k*π

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