Find the Exact Value csc(85)

Problem

csc(85^∘)

Solution

1. Identify the expression as the cosecant of an angle in degrees. The value csc(85^∘) is equivalent to 1/sin(85^∘)

2. Recognize that 85^∘ is not a standard reference angle (like 30^∘ 45^∘ or 60^∘. To find an exact radical expression, we can use the angle sum formula 85^∘=45^∘+40^∘ or 85^∘=60^∘+25^∘ but these do not lead to standard radicals easily.

3. Apply the half-angle or sum/difference identities if possible. However, 85^∘ is typically expressed in exact form using nested radicals via the sum 85^∘=45^∘+40^∘ or by relating it to 5^∘ (90^∘−5^∘.

4. Determine the exact value using the sine addition formula for sin(85^∘)=sin(45^∘+40^∘) or sin(90^∘−5^∘)=cos(5^∘) The exact value for cos(5^∘) is:

cos(5^∘)=√(,2)/4*(√(,3+√(,5))+√(,5−√(,5)))

5. Invert the sine value to find the cosecant. Given the complexity of 85^∘ the most direct "exact" representation without further simplification of nested radicals is the reciprocal of the sine.

Final Answer

csc(85^∘)=1/sin(85^∘)

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