Find the Exact Value csc((5pi)/3)

Problem

csc((5*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 reference angle for (5*π)/3 by finding its distance from the nearest x-axis value, 2*π

Reference Angle=2*π−(5*π)/3

Reference Angle=π/3

3. Determine the sign of the sine function in the fourth quadrant. Since (5*π)/3 is in Quadrant IV, the sine value is negative.

sin((5*π)/3)=−sin(π/3)

4. Substitute the known exact value for sin(π/3) which is √(,3)/2

sin((5*π)/3)=−√(,3)/2

5. Calculate the cosecant by taking the reciprocal of the sine value.

csc((5*π)/3)=1/(−√(,3)/2)

csc((5*π)/3)=−2/√(,3)

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

csc((5*π)/3)=−(2√(,3))/3

Final Answer

csc((5*π)/3)=−(2√(,3))/3

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