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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