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

Problem

csc(x)=2/√(,3)

Solution

1. Rewrite the equation using the reciprocal identity csc(x)=1/sin(x)

1/sin(x)=2/√(,3)

2. Invert both sides of the equation to solve for sin(x)

sin(x)=√(,3)/2

3. Identify the reference angle in the first quadrant where the sine value is √(,3)/2

x=π/3

4. Determine the second solution within the interval [0,2*π) by using the property that sine is positive in the second quadrant.

x=π−π/3

x=(2*π)/3

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

x=π/3+2*π*n

x=(2*π)/3+2*π*n

Final Answer

csc(x)=2/√(,3)⇒x=π/3+2*π*n,(2*π)/3+2*π*n

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