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

Problem

sin(x)=−√(,3)/2

Solution

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

θ=π/3

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 specific solutions within the interval [0,2*π) using the reference angle.

x=π+π/3=(4*π)/3

x=2*π−π/3=(5*π)/3

4. Generalize the solution to include all possible angles by adding integer multiples of the period 2*π where n is any integer.

x=(4*π)/3+2*n*π

x=(5*π)/3+2*n*π

Final Answer

sin(x)=−√(,3)/2⇒x=(4*π)/3+2*n*π,(5*π)/3+2*n*π

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