Solve for x csc(x)=(2 square root of 3)/3

Problem

csc(x)=(2√(,3))/3

Solution

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

1/sin(x)=(2√(,3))/3

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

sin(x)=3/(2√(,3))

3. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

sin(x)=(3√(,3))/(2⋅3)

4. Simplify the fraction by canceling the common factor of 3

sin(x)=√(,3)/2

5. Identify the reference angle in the first quadrant where the sine value is √(,3)/2 which is x=π/3 or 60

x=π/3+2*n*π

6. Determine the second solution within one period by using the identity sin(x)=sin(π−x) which gives x=π−π/3=(2*π)/3

x=(2*π)/3+2*n*π

Final Answer

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

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