Find the Exact Value csc(5)

Problem

csc(5)

Solution

1. Identify the expression as the cosecant of 5 degrees.

2. Relate the cosecant function to the sine function using the reciprocal identity.

csc(5)=1/sin(5)

3. Apply the half-angle formula for sine, where sin(θ/2)=√(,(1−cos(θ))/2) using θ=10

sin(5)=√(,(1−cos(10))/2)

4. Determine the exact value of cos(10) using the triple-angle identity for cos(30) or other radical forms, which results in a complex nested radical.

5. Express the final result in its exact trigonometric form as the reciprocal of the sine of 5 degrees.

Final Answer

csc(5)=1/sin(5)

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