Solve for x square root of 3csc(x)-2=0

Problem

√(,3)*csc(x)−2=0

Solution

1. Isolate the cosecant term by adding 2 to both sides of the equation.

√(,3)*csc(x)=2

2. Divide both sides by √(,3) to solve for csc(x)

csc(x)=2/√(,3)

3. Rewrite the equation in terms of the sine function using the reciprocal identity csc(x)=1/sin(x)

sin(x)=√(,3)/2

4. Identify the reference angle in the first quadrant where the sine value is √(,3)/2 which is π/3 or 60

x=π/3

5. Determine all solutions within the standard interval [0,2*π) by finding where sine is positive (Quadrants I and II).

x=π/3

x=π−π/3=(2*π)/3

6. Generalize the solution 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

√(,3)*csc(x)−2=0⇒x=π/3+2*π*n,(2*π)/3+2*π*n

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