Find the Exact Value tan(54)

Problem

tan(54)

Solution

1. Identify the angle as 54 which is equivalent to (3*π)/10 radians.

2. Use the relationship tan(54)=cot(36) based on the co-function identity tan(90−θ)=cot(θ)

3. Recall the exact value for sin(18)=(√(,5)−1)/4 which allows for the derivation of other trigonometric values for multiples of 18

4. Apply the formula for tan(54) derived from the geometry of a regular pentagon or half-angle identities, which results in √(,1+2/√(,5))

5. Simplify the expression into the standard radical form √(,1+(2√(,5))/5) or the more common nested radical form.

Final Answer

tan(54)=√(,1+2/√(,5))

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