Find the Exact Value tan(65)

Problem

tan(65^∘)

Solution

1. Identify the angle as a sum of two special angles. We can write 65^∘ as 45^∘+20^∘ or 35^∘+30^∘ but since 65^∘ is not a standard multiple of 15^∘ or 22.5^∘ the exact value is typically expressed using the tangent addition formula or radical forms involving roots of polynomials.

2. Apply the tangent addition formula tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B)) if the components are known.

3. Recognize that 65^∘ is the complement of 25^∘ so tan(65^∘)=cot(25^∘)

4. Determine the exact radical form. The value tan(65^∘) does not have a simple expression using only square roots (like tan(75^∘)=2+√(,3). It involves roots of higher-degree polynomials or complex trigonometric constants.

5. State the value in its simplest exact trigonometric form or its known radical approximation if applicable. For most contexts, the exact value is simply tan(65^∘) or cot(25^∘)

Final Answer

tan(65^∘)=tan(65^∘)

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