Find the Exact Value csc(5/3)

Problem

csc((5*π)/3)

Solution

1. Identify the angle in the unit circle. The angle (5*π)/3 is in the fourth quadrant because it is between (3*π)/2 and 2*π

2. Find the reference angle. The reference angle θ′ is calculated by subtracting the angle from 2*π

θ′=2*π−(5*π)/3

θ′=π/3

3. Determine the cosecant value using the reciprocal identity. The cosecant function is the reciprocal of the sine function.

csc((5*π)/3)=1/sin((5*π)/3)

4. Evaluate the sine of the angle. In the fourth quadrant, sine is negative.

sin((5*π)/3)=−sin(π/3)

sin((5*π)/3)=−√(,3)/2

5. Calculate the reciprocal to find the cosecant.

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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