Find the Exact Value csc(65)

Problem

csc(65^∘)

Solution

1. Identify the function and the angle. The task is to find the exact value of the cosecant of 65^∘

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

csc(65^∘)=1/sin(65^∘)

3. Express the angle 65^∘ as a sum or difference of standard angles if possible. Since 65^∘ is not a standard angle (like 30^∘,45^∘,60^∘ or a simple combination of them that results in a standard radical form, the exact value is typically represented in terms of sine or cosine.

4. Determine if a radical form exists. While 65^∘ can be written as 45^∘+20^∘ or 90^∘−25^∘ these do not lead to a standard algebraic constant involving simple square roots.

5. Conclude that without further context or specific identities for 5^∘ increments, the exact value is the reciprocal of the sine of the angle.

Final Answer

csc(65^∘)=1/sin(65^∘)

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