Find the Exact Value csc(80)

Problem

csc(80^∘)

Solution

1. Identify the trigonometric function and the angle. The task is to find the exact value of the cosecant of 80^∘

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

csc(80^∘)=1/sin(80^∘)

3. Express the angle in terms of a known identity. Since 80^∘ is not a standard reference angle, we can express it as 90^∘−10^∘ to use cofunction identities.

sin(80^∘)=sin(90^∘−10^∘)=cos(10^∘)

4. Determine if a simpler radical form exists. The angle 80^∘ (or (4*π)/9 radians) does not result in a simple radical expression using standard square roots. Its exact value involves cube roots of complex numbers via the triple-angle formula for cos(3*θ) where θ=10^∘ or 80^∘

5. Conclude that without further context for a specific nested radical form, the simplest exact trigonometric representation is the reciprocal of the sine or the secant of the complement.

Final Answer

csc(80^∘)=1/sin(80^∘)

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