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

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 the second solution within the interval [0,2*π) by using the property that sine is positive in the second quadrant, calculated as π−reference angle

x=π−π/6=(5*π)/6

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

Want more problems? Check here!