Find the Exact Value csc((4pi)/3)

Problem

csc((4*π)/3)

Solution

1. Identify the reference angle for (4*π)/3 by determining its distance from the nearest x-axis.

θ′=(4*π)/3−π

θ′=π/3

2. Determine the quadrant of the angle (4*π)/3 Since π<(4*π)/3<(3*π)/2 the angle lies in Quadrant III.

3. Determine the sign of the cosecant function in Quadrant III. Since csc(θ)=1/sin(θ) and sine is negative in Quadrant III, the result will be negative.

4. Evaluate the cosecant of the reference angle π/3

csc(π/3)=1/sin(π/3)

csc(π/3)=1/√(,3)/2

csc(π/3)=2/√(,3)

5. Apply the sign from step 3 to the value found in step 4.

csc((4*π)/3)=−2/√(,3)

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

csc((4*π)/3)=−(2√(,3))/3

Final Answer

csc((4*π)/3)=−(2√(,3))/3

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