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

Problem

csc(x)=√(,2)

Solution

1. Rewrite the equation using the reciprocal identity csc(x)=1/sin(x)

1/sin(x)=√(,2)

2. Isolate the sine function by taking the reciprocal of both sides.

sin(x)=1/√(,2)

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

sin(x)=√(,2)/2

4. Determine the reference angle in the first quadrant where the sine value is √(,2)/2

x=π/4

5. Identify the second solution within the interval [0,2*π) by using the property that sine is positive in the second quadrant.

x=π−π/4=(3*π)/4

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

x=π/4+2*n*π

x=(3*π)/4+2*n*π

Final Answer

csc(x)=√(,2)⇒x=π/4+2*n*π,(3*π)/4+2*n*π

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