Solve for x csc(x)=2

Problem

csc(x)=2

Solution

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

1/sin(x)=2

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

sin(x)=1/2

3. Identify the reference angle in the first quadrant where the sine value is 1/2

x=π/6

4. Determine all solutions within the standard interval [0,2*π) by finding the second quadrant angle where sine is also positive.

x=π−π/6=(5*π)/6

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

x=π/6+2*n*π

x=(5*π)/6+2*n*π

Final Answer

csc(x)=2⇒x=π/6+2*n*π,(5*π)/6+2*n*π

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