Solve for x csc(x)-2=0

Problem

csc(x)−2=0

Solution

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

csc(x)=2

2. Rewrite the cosecant function in terms of its reciprocal, the sine function, using the identity csc(x)=1/sin(x)

1/sin(x)=2

3. Solve for sin(x) by taking the reciprocal of both sides.

sin(x)=1/2

4. Determine the reference angle by finding the value in the first quadrant where the sine is 1/2

x=π/6

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

x=π/6

x=π−π/6=(5*π)/6

6. Generalize the solution to include all possible rotations by adding multiples of 2*π*n where n is an integer.

x=π/6+2*π*n

x=(5*π)/6+2*π*n

Final Answer

csc(x)−2=0⇒x=π/6+2*π*n,(5*π)/6+2*π*n

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