Find the Exact Value 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. Substitute the given angle into the identity.

csc((2*π)/3)=1/sin((2*π)/3)

3. Determine the sine of the angle. Since (2*π)/3 is in the second quadrant and has a reference angle of π/3 the sine value is positive.

sin((2*π)/3)=√(,3)/2

4. Calculate the reciprocal of the sine value.

csc((2*π)/3)=1/√(,3)/2

5. Simplify the fraction by multiplying by the reciprocal.

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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