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

Problem

csc((7*π)/3)

Solution

1. Identify the angle and determine its coterminal equivalent within the standard interval [0,2*π] by subtracting 2*π

(7*π)/3−2*π=(7*π)/3−(6*π)/3

(7*π)/3−(6*π)/3=π/3

2. Apply the definition of the cosecant function, which is the reciprocal of the sine function.

csc(π/3)=1/sin(π/3)

3. Substitute the known exact value for sin(π/3)

sin(π/3)=√(,3)/2

4. Simplify the fraction by taking the reciprocal.

1/√(,3)/2=2/√(,3)

5. 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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