Find the Exact Value csc(pi/3)

Problem

csc(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 π/3 into the identity.

csc(π/3)=1/sin(π/3)

3. Determine the exact value of sin(π/3) using the unit circle or a special right triangle (60.

sin(π/3)=√(,3)/2

4. Simplify the expression by taking the reciprocal of the sine value.

csc(π/3)=1/√(,3)/2

csc(π/3)=2/√(,3)

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

csc(π/3)=(2⋅√(,3))/(√(,3)⋅√(,3))

csc(π/3)=(2√(,3))/3

Final Answer

csc(π/3)=(2√(,3))/3

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