Solve for x csc(x)=- square root of 2

Problem

csc(x)=−√(,2)

Solution

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

1/sin(x)=−√(,2)

2. Isolate sin(x) by taking the reciprocal of both sides of the equation.

sin(x)=−1/√(,2)

3. Rationalize the denominator to identify the standard value on the unit circle.

sin(x)=−√(,2)/2

4. Identify the reference angle by finding the value in the first quadrant where sin(θ)=√(,2)/2

(x_ref)=π/4

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

x=π+π/4=(5*π)/4

x=2*π−π/4=(7*π)/4

6. Generalize the solution by adding integer multiples of the period 2*π where n is an integer.

x=(5*π)/4+2*n*π

x=(7*π)/4+2*n*π

Final Answer

csc(x)=−√(,2)⇒x=(5*π)/4+2*n*π,(7*π)/4+2*n*π

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