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

Problem

√(,3)*csc(x)+2=0

Solution

1. Isolate the cosecant term by subtracting 2 from 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 by finding the value in the first quadrant where sin(x)=√(,3)/2 which is π/3

(x_ref)=π/3

5. Determine the quadrants where the sine function is negative, which are Quadrant III and Quadrant IV.

x=π+π/3=(4*π)/3

x=2*π−π/3=(5*π)/3

6. Generalize the solution by adding 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

√(,3)*csc(x)+2=0⇒x=(4*π)/3+2*n*π,(5*π)/3+2*n*π

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