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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