Find the Exact Value csc(-(7pi)/6)

Problem

csc(−(7*π)/6)

Solution

1. Apply the odd function property of the cosecant function, which states csc(−θ)=−csc(θ)

csc(−(7*π)/6)=−csc((7*π)/6)

2. Identify the reference angle for (7*π)/6 Since (7*π)/6=π+π/6 the angle is in the third quadrant and the reference angle is π/6

Reference Angle=π/6

3. Determine the sign of the cosecant function in the third quadrant. Since sine (and therefore cosecant) is negative in the third quadrant, csc((7*π)/6)=−csc(π/6)

−csc((7*π)/6)=−(−csc(π/6))

4. Simplify the signs to show that the expression is equivalent to the cosecant of the reference angle.

−(−csc(π/6))=csc(π/6)

5. Evaluate the exact value using the reciprocal identity csc(θ)=1/sin(θ) Since sin(π/6)=1/2 the cosecant is the reciprocal.

csc(π/6)=2

Final Answer

csc(−(7*π)/6)=2

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