Find the Exact Value csc((7pi)/3)

Problem

csc((7*π)/3)

Solution

1. Find the coterminal angle by subtracting 2*π from the given angle to find an equivalent angle within the interval [0,2*π)

(7*π)/3−2*π=(7*π)/3−(6*π)/3

(7*π)/3−(6*π)/3=π/3

2. Identify the reciprocal identity for the cosecant function.

csc(θ)=1/sin(θ)

3. Substitute the angle into the identity.

csc(π/3)=1/sin(π/3)

4. Evaluate the sine of the reference angle using the unit circle.

sin(π/3)=√(,3)/2

5. Simplify the fraction by taking the reciprocal of the sine value.

csc(π/3)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

csc((7*π)/3)=(2√(,3))/3

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