Solve for x csc(x)=-2

Problem

csc(x)=−2

Solution

1. Rewrite the equation 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.

sin(x)=−1/2

3. Identify the reference angle by finding the value in the first quadrant where sin(θ)=1/2

θ=π/6

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

Quadrant III: *x=π+π/6=(7*π)/6

Quadrant IV: *x=2*π−π/6=(11*π)/6

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

x=(7*π)/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

csc(x)=−2⇒x=(7*π)/6+2*n*π,(11*π)/6+2*n*π

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