Find the Value Using the Unit Circle csc((7pi)/6)

Problem

csc((7*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. Determine the reference angle for (7*π)/6 by finding its distance from the x-axis.

(θ_ref)=(7*π)/6−π=π/6

3. Locate the quadrant of the angle (7*π)/6 Since π<(7*π)/6<(3*π)/2 the angle lies in Quadrant III.

4. Evaluate the sine of the angle. In Quadrant III, the sine value is negative.

sin((7*π)/6)=−sin(π/6)

sin((7*π)/6)=−1/2

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

csc((7*π)/6)=1/(−1/2)

csc((7*π)/6)=−2

Final Answer

csc((7*π)/6)=−2

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