Find the Exact Value csc((7pi)/2)

Problem

csc((7*π)/2)

Solution

1. Identify the definition of the cosecant function in terms of the sine function.

csc(θ)=1/sin(θ)

2. Substitute the given angle into the identity.

csc((7*π)/2)=1/sin((7*π)/2)

3. Find a coterminal angle for (7*π)/2 by subtracting 2*π (one full rotation) from the angle.

(7*π)/2−2*π=(7*π)/2−(4*π)/2

(7*π)/2−(4*π)/2=(3*π)/2

4. Evaluate the sine of the coterminal angle using the unit circle.

sin((3*π)/2)=−1

5. Calculate the final value by taking the reciprocal.

1/(−1)=−1

Final Answer

csc((7*π)/2)=−1

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