Find the Value Using the Unit Circle csc((2pi)/3)

Problem

csc((2*π)/3)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(θ)=1/sin(θ)

2. Determine the coordinates of the angle (2*π)/3 on the unit circle.

θ=(2*π)/3

3. Find the ycoordinate at (2*π)/3 which represents the value of sin((2*π)/3)

sin((2*π)/3)=√(,3)/2

4. Substitute the sine value into the reciprocal identity to find the cosecant.

csc((2*π)/3)=1/√(,3)/2

5. Simplify the expression by multiplying by the reciprocal of the denominator.

csc((2*π)/3)=2/√(,3)

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

csc((2*π)/3)=(2√(,3))/3

Final Answer

csc((2*π)/3)=(2√(,3))/3

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