Find the Exact Value csc((11pi)/3)

Problem

csc((11*π)/3)

Solution

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

(11*π)/3−2*π=(11*π)/3−(6*π)/3

(11*π)/3−(6*π)/3=(5*π)/3

2. Identify the quadrant for the angle (5*π)/3 Since (3*π)/2<(5*π)/3<2*π the angle is in Quadrant IV.

3. Determine the reference angle θ′ for an angle in Quadrant IV using the formula θ′=2*π−θ

θ′=2*π−(5*π)/3

θ′=π/3

4. Apply the reciprocal identity for cosecant, which is csc(x)=1/sin(x)

csc((5*π)/3)=1/sin((5*π)/3)

5. Evaluate the sine of the angle. In Quadrant IV, sine is negative.

sin((5*π)/3)=−sin(π/3)

sin((5*π)/3)=−√(,3)/2

6. Calculate the reciprocal to find the cosecant value.

csc((11*π)/3)=1/(−√(,3)/2)

csc((11*π)/3)=−2/√(,3)

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

csc((11*π)/3)=−(2√(,3))/3

Final Answer

csc((11*π)/3)=−(2√(,3))/3

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