Find the Exact Value csc((5pi)/6)

Problem

csc((5*π)/6)

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 into the identity.

csc((5*π)/6)=1/sin((5*π)/6)

3. Determine the value of the sine function at the given angle. The angle (5*π)/6 is in the second quadrant, where sine is positive, and its reference angle is π−(5*π)/6=π/6

sin((5*π)/6)=sin(π/6)

4. Evaluate the sine of the reference angle using the unit circle or special right triangles.

sin(π/6)=1/2

5. Calculate the final value by taking the reciprocal.

1/(1/2)=2

Final Answer

csc((5*π)/6)=2

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